2.1. From standard to sparse regular variation

We start from Equation (1.1), which we rephrase as follows:

(2.1) \begin{equation}{\mathbb{P}} ( (|{\textbf{X}}|/t, {\textbf{X}} / |{\textbf{X}}|) \in \cdot \mid |{\textbf{X}}| > t ) \stackrel{w}{\to} {\mathbb{P}}( (Y, {\boldsymbol{\Theta}}) \in \cdot )\, , \quad t \to \infty\, ,\end{equation}

where Y is a Pareto( $\alpha$ )-distributed random variable independent of ${\boldsymbol{\Theta}}$ . We call the random vector ${\boldsymbol{\Theta}}$ the spectral vector, its distribution being the spectral measure. If the convergence (2.1) holds we write ${\textbf{X}} \in \mathrm{RV}(\alpha, {\boldsymbol{\Theta}})$ . In many cases the spectral measure is sparse; that is, it places mass on some lower-dimensional subspaces. The self-normalized vector ${\textbf{X}}/|{\textbf{X}}|$ then fails to estimate the spectral vector ${\boldsymbol{\Theta}}$ on such subsets.

In order to better capture this sparsity, we replace the quantity ${\textbf{X}}/|{\textbf{X}}|$ by the vector $\pi_1({\textbf{X}}/t)$ , where $\pi_z({\textbf{v}})$ denotes the Euclidean projection of the vector ${\textbf{v}}$ in $\mathbb{R}^d_+$ onto the positive sphere $\mathbb S^{d-1}_+(z) \,:\!=\, \{{\textbf{x}} \in \mathbb{R}^d_+ \,:\, x_1 + \ldots + x_d = z\}$ , $z> 0$ . The projected vector $\pi_z({\textbf{v}})$ is defined as the unique solution of the following optimization problem (see [Reference Duchi, Shalev-Shwartz, Singer and Chandra10] and the references therein):

(2.2) \begin{equation} \underset{{\textbf{w}}}{\mbox{minimize }} \frac{1}{2} |{\textbf{w}} - {\textbf{v}}|_2^2 \quad \mbox{s.t.} \quad |{\textbf{w}}|_1 = z\, .\end{equation}

The Euclidean projection onto the positive sphere $\mathbb S^{d-1}_+(z)$ is then defined as the application

\begin{equation*}\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}l}\pi_z & : & \mathbb{R}^d_+ & \to & \mathbb S^{d-1}_+(z) \\[5pt] & & {\textbf{v}} & \mapsto & {\textbf{w}} = ({\textbf{v}}-\lambda_{{\textbf{v}},z})_+\, ,\end{array}\end{equation*}

where $\lambda_{{\textbf{v}}, z}$ is the unique constant satisfying the relation $\sum_{i=1}^d (v_i - \lambda_{{\textbf{v}},z})_+ =z$ . Several algorithms which compute $\pi_z({\textbf{v}})$ have been introduced [Reference Duchi, Shalev-Shwartz, Singer and Chandra10, Reference Condat8]. We present two of them in Appendix A: first one that gives an intuitive way to compute $\pi_z({\textbf{v}})$ , and then one based on a median-search procedure whose expected time complexity is O(d).

Figure 1.

The Euclidean projection onto the simplex $\mathbb S^{1}_+$ .

The projection satisfies the relation $ \pi_z({\textbf{v}}) = z\pi_1({\textbf{v}}/z) $ for all ${\textbf{v}} \in \mathbb{R}^d_+$ and $z > 0$ . This is why we mainly focus on the projection $\pi_1$ onto the simplex $\mathbb S^{d-1}_+$ . In this case we write $\pi$ as shorthand for $\pi_1$ . An illustration of $\pi$ for $d=2$ is given in Figure 1. We list below some straightforward results satisfied by the projection:

1. P1. The projection preserves the order of the coordinates: if $v_{\sigma(1)} \geq \ldots \geq v_{\sigma(d)}$ for a permutation $\sigma$ , then $\pi({\textbf{v}})_{\sigma(1)} \geq \ldots \geq \pi({\textbf{v}})_{\sigma(d)}$ for the same permutation.

2. P2. If $\pi({\textbf{v}})_j > 0$ , then $v_j > 0$ . Equivalently, $v_j = 0$ implies $\pi({\textbf{v}})_j = 0$ .

3. P3. The projection $\pi$ is continuous, as is every projection on a convex, closed set in a Hilbert space.

The substitution of the self-normalized vector ${\textbf{X}}/|{\textbf{X}}|$ by the projected vector $\pi({\textbf{X}}/t)$ motivates the following definition.

Definition 1. (Sparse regular variation.) A random vector ${\textbf{X}} \in \mathbb{R}^d_+$ is sparsely regularly varying if there exist a random vector ${\textbf{Z}}$ defined on the simplex $\mathbb{S}^{d-1}_+$ and a nondegenerate random variable Y such that

(2.3) \begin{equation}{\mathbb{P}} ( ( |{\textbf{X}}| / t , \pi ({\textbf{X}} / t ) ) \in \cdot \mid |{\textbf{X}}| > t ) \overset{w}{\to} {\mathbb{P}}((Y, {\textbf{Z}}) \in \cdot)\, , \quad t \rightarrow \infty\,.\end{equation}

If $\textbf{X}$ satisfies (2.3), we write ${\textbf{X}} \in \mathrm{SRV}(\alpha, {\textbf{Z}})$ .

In this case, standard results on regularly varying random variables state that there exists $\alpha > 0$ such that Y follows a Pareto distribution with parameter $\alpha$ . The continuity of $\pi$ ensures that regular variation with limit $(Y, {\boldsymbol{\Theta}})$ implies sparse regular variation with limit $(Y, \pi(Y {\boldsymbol{\Theta}}))$ . In contrast to the limit in (2.1), here we lose independence between the radial component Y and the angular component ${\textbf{Z}}$ of the limit. The dependence relation between the two components is given by the following proposition.

Proposition 1. If ${\textbf{X}} \in \mathrm{SRV}(\alpha, {\textbf{Z}})$ , then for all $r \geq 1$ we have

\begin{equation*}{\mathbb{P}}({\textbf{Z}} \in \cdot \mid Y > r) \overset{d}{=} {\mathbb{P}}( \pi(r {\textbf{Z}}) \in \cdot) \,.\end{equation*}

For $\beta \in {{\mathcal{P}}}_d^*$ we denote by ${\textbf{e}}(\beta)$ the vector with 1 in position j if $j \in \beta$ and 0 otherwise. Then the vector ${\textbf{e}}(\beta)/|\beta|$ belongs to the simplex. We consider the following class of discrete distributions on the simplex:

(2.4) \begin{equation} \sum_{ \beta \in {{\mathcal{P}}}_d^* } p(\beta) \, \delta_{{\textbf{e}}(\beta) / |\beta|}\, ,\end{equation}

where $(p(\beta))_\beta$ is a $(2^d-1)$ -vector with nonnegative components summing to 1. This is the device developed by [Reference Segers32]. The family of distributions (2.4) is stable under multiplication by a positive random variable and Euclidean projection onto the simplex. Hence, if ${\boldsymbol{\Theta}}$ has a distribution of type (2.4), then ${\textbf{Z}} = {\boldsymbol{\Theta}}$ almost surely (a.s.). The following corollary states that distributions of this kind are the only possible discrete distributions for ${\textbf{Z}}$ .

The family of distributions given in (2.4) forms an accurate model for the angular vector ${\textbf{Z}}$ . Indeed, the distributions of this class place mass on some particular points of the simplex on which extremes values often concentrate in practice. They include the case of complete dependence, which corresponds to the case $p( \{1, \ldots, d \} ) = 1$ , and the case of asymptotic independence, which corresponds to the case $p(\{j\}) = 1/d$ for all $j = 1, \ldots, d$ (see also Example 2).

2.2. Main result

We establish in this section some relations between the vectors ${\boldsymbol{\Theta}}$ and ${\textbf{Z}}$ . We start by introducing some notation which will be useful throughout the article. We consider the sets

\begin{equation*}A_{\textbf{x}} = \{{\textbf{u}} \in \mathbb S^{d-1}_+ \,:\, {\textbf{u}} \geq {\textbf{x}}\}\, , \quad {\textbf{x}} \in B_+^d(0,1)\, ,\end{equation*}

\begin{equation*}{{\mathcal{X}}}_\beta = \{{\textbf{x}} \in B_+^d(0,1) \,:\, {\textbf{x}}_\beta > {\textbf{0}}_\beta, \, {\textbf{x}}_{\beta^c} = {\textbf{0}}_{\beta^c} \}\, , \quad \beta \in {{\mathcal{P}}}_d^*\, ,\end{equation*}

\begin{equation*}{{\mathcal{X}}}_\beta^0 = \{{\textbf{x}} \in B_+^d(0,1) \,:\, {\textbf{x}}_{\beta^c} = {\textbf{0}}_{\beta^c} \}\, , \quad \beta \in {{\mathcal{P}}}_d^*\, ,\end{equation*}

and we define $\lambda_\beta$ as the Lebesgue measure on the set ${{\mathcal{X}}}_\beta$ . For $\beta, \gamma \in {{\mathcal{P}}}_d^*$ such that $\gamma \supset \beta$ we also consider the functions

\begin{equation*}G_\beta({\textbf{x}}) = {\mathbb{P}}( {\textbf{Z}}_\beta > {\textbf{x}}_\beta, \, {\textbf{Z}}_{\beta^c} \leq {\textbf{x}}_{\beta^c})\, , \quad {\textbf{x}}_\beta \in B^{|\beta|}_+(0,1)\, , \quad {\textbf{x}}_{\beta^c} \in B^{|\beta^c|}_+(0,1) \, ,\end{equation*}

and

\begin{equation*}H_{\beta, \gamma}({\textbf{u}}, v, w)= {\mathbb{P}} \Big( \phi_\gamma({\textbf{Z}})_\beta \geq {\textbf{u}}, \,\min_{j \in \gamma \setminus \beta} \phi_\gamma({\textbf{Z}})_j > v, \,\max_{j \in \gamma^c} \phi_\gamma({\textbf{Z}})_j \leq w \Big)\, ,\end{equation*}

for ${\textbf{u}} \in B_+^{|\beta|}(0,1)$ , $v, w \in [0,1]$ , and with $\phi_\gamma \,:\, \mathbb S^{d-1}_+ \to \mathbb{R}^d_+, \, {\textbf{u}} \mapsto \phi_\gamma({\textbf{u}})$ defined as

\begin{equation*}\phi_\gamma({\textbf{u}})_j=\left\{\begin{array}{ll}u_j + \frac{ |{\textbf{u}}_{\gamma^c}| }{ |\gamma| }\, \quad \text{for }j \in \gamma\, ,\\[6pt] u_j + \frac{ |{\textbf{u}}_{\gamma^c\setminus\{j\}}| }{ |\gamma|+1 }\, \quad \text{for }j \in \gamma^c\, ;\end{array}\right.\end{equation*}

see Lemma 3 for more insights on this function.

We consider the following assumption:

1. (A) For all $\beta, \gamma \in {{\mathcal{P}}}_d^*$ such that $\gamma \supset \beta$ and for $\lambda_\beta$ -almost every ${\textbf{x}} \in {{\mathcal{X}}}_\beta$ the function $H_{\beta, \gamma}$ is continuously differentiable in $({\textbf{x}}_\beta,0,0)$ .

We are now able to state the main result of this paper.

Equation (2.5) gives a relation between the distribution of ${\textbf{Z}}$ and that of ${\boldsymbol{\Theta}}$ . While its complexity makes it difficult to use in full generality, specific choices for ${\textbf{x}}$ lead to useful results. A convenient special case is the one with $\beta = \{1, \ldots, d\}$ and ${\textbf{x}} < 1/d$ , which provides the relation

\begin{equation*}G_{\textbf{Z}}({\textbf{x}})= {\mathbb{E}} \bigg[ \bigg( 1 - \max_{1 \leq j \leq d} \bigg(\frac{1/d - \Theta_j}{1/d-x_j} \bigg)^\alpha\bigg)_+ \bigg] \, .\end{equation*}

In particular, for ${\textbf{x}} = {\textbf{0}}$ we obtain

(2.7) \begin{equation} G_{\textbf{Z}}({\textbf{0}}) = 1 - {\mathbb{E}} \Big[ \max_{1\leq j \leq d } (1-d\Theta_j)^\alpha \Big] \, .\end{equation}

Thus, the probability that ${\textbf{Z}}$ has a null component is

(2.8) \begin{equation} {\mathbb{P}}(Z_j=0 \text{ for some } j \leq d) = {\mathbb{E}} \Big[ \max_{1 \leq j \leq d } (1-d \Theta_j)^\alpha \Big]\,.\end{equation}

This quantity is null if and only if for all $j = 1, \ldots, d$ , $\Theta_j=1/d$ a.s., and is equal to 1 if and only if $\min_{1 \leq j \leq d}\Theta_j=0$ a.s. This implies that the new angular vector ${\textbf{Z}}$ is more likely to be sparse in the sense of Remark 1. In particular, all of the usual spectral models on ${\boldsymbol{\Theta}}$ that are not supported on the axis are unsuitable for ${\textbf{Z}}$ . More insights into the sparsity of the vector ${\textbf{Z}}$ are given in Section 3.

Example 1. We consider a spectral vector ${\boldsymbol{\Theta}}$ in $\mathbb S^1_+$ with a first component $\Theta_1$ uniformly distributed (and then $\Theta_2 = 1 - \Theta_1$ is also uniformly distributed). This fits into the framework of Remark 2. We also assume $\alpha = 1$ for simplicity. The probability that ${\textbf{Z}}$ belongs to the first axis is equal to

\begin{equation*}{\mathbb{P}}(Z_1 = 1) = {\mathbb{P}}(Y \Theta_1 - Y \Theta_2 \geq 1) = {\mathbb{P}}(2 \Theta_1 - 1 \geq 1/Y)\end{equation*}

(see Lemma 3). Since the random variable $1/Y$ follows a uniform distribution on (0,1) and is independent of ${\boldsymbol{\Theta}}$ we obtain

(2.9) \begin{equation}{\mathbb{P}}(Z_1 = 1) = \int_0^1 {\mathbb{P}}(2 \Theta_1 - 1 \geq u) \, \mathrm d u = \int_0^1 {\mathbb{P}} \Big(\Theta_1 \geq \frac{u+1}{2} \Big) \, \mathrm d u = \int_0^1 \frac{1-u}{2} \, \mathrm d u = 1/4 \, . \end{equation}

Furthermore, if $x \in (0,1)$ , Lemma 3 implies that

\begin{align*} {\mathbb{P}}(0 < Z_1 \leq x) &= {\mathbb{P}}(0 < Y \Theta_1 -(Y-1)/2 \leq x)\\ &= \int_0^1 {\mathbb{P}}\big[ (1-u)/2 < \Theta_1 \leq (1 +(2x-1)u) / 2 \big] \, \mathrm d u = x/2\,.\end{align*}

The distribution of $Z_1$ is thus given by $\delta_0/4 + \delta_1/4 + U/2$ , where U denotes a uniform distribution on (0,1) and $\delta_a$ a Dirac measure at the point a. We check in Appendix B that this vector ${\textbf{Z}}$ satisfies Equations (2.5) and (2.6).

3.1. Sparsity in extremes

This section tackles the issue of detecting extremal directions for a regularly varying random vector ${\textbf{X}}$ . In such a context it is helpful to partition the underlying space, in our case $\mathbb S ^{d-1}_+$ , into understandable subsets [Reference Chautru4, Reference Goix, Sabourin and ClémenÇon18, Reference Simpson, Wadsworth and Tawn34]. In this article we consider the subsets

(3.1) \begin{equation} C_\beta = \big \{ {\textbf{x}} \in \mathbb S^d_+ \,:\, x_i > 0 \text{ for } i \in \beta, \, x_i = 0 \text{ for } i \notin \beta \big\} \, , \quad \beta \in {{\mathcal{P}}}_d^*\, ,\end{equation}

which form a partition of the simplex. An illustration of these subsets in dimension 3 is given in Figure 2. This partition is helpful for studying the tail structure of ${\textbf{X}}$ . Indeed, for $\beta \in {{\mathcal{P}}}_d^*$ the inequality ${\mathbb{P}}({\boldsymbol{\Theta}} \in C_\beta) > 0$ means that one is likely to simultaneously observe large values in the directions $i \in \beta$ and small values in the directions $i \in \beta^c$ . Then identifying the subsets $C_\beta$ which concentrate the mass of the spectral measure allows us to bring out clusters of coordinates which can be simultaneously large.

Figure 2.

The subsets $C_\beta$ in dimension 3 for the $\ell^1$ -norm. The subsets $C_{ \{1\} }$ , $C_{ \{2\} }$ , and $C_{ \{3\} }$ correspond to the unit vectors ${\textbf{e}}_1$ , ${\textbf{e}}_2$ , and ${\textbf{e}}_3$ , respectively. The dashed lines indicate the subsets $C_{ \{1, 2\} }$ , $C_{ \{1, 3\} }$ , and $C_{ \{2, 3\} }$ . Finally, the subset $C_{\{1, 2, 3 \}}$ corresponds to the interior of the simplex.

As for any low-dimensional subspaces, topological issues may arise for the subsets $C_\beta$ in the convergence (2.1). Indeed, for $\beta \neq \{1, \ldots, d\}$ , the subset $C_\beta$ is included in its boundary (with respect to the topology of the simplex), and the convergence (2.1) fails for such a set. This kind of problem appears because the spectral measure may put mass on low-dimensional subspaces, while the data generally do not concentrate on such subspaces. This issue can be circumvented with sparse regular variation. We replace the study of the sets $\{ {\textbf{x}} \in \mathbb{R}^d_+ \,:\, |{\textbf{x}}| > 1, \, {\textbf{x}} / |{\textbf{x}}| \in C_\beta\}$ by the study of the sets $\{ {\textbf{x}} \in \mathbb{R}^d_+ \,:\, |{\textbf{x}}| > 1, \, \pi({\textbf{x}} / t) \in C_\beta\}$ , which enjoy better topological properties.

In this context, many other sets have recently been proposed. [Reference Goix, Sabourin and ClémenÇon17] defined the truncated $\epsilon$ -cones as

\begin{equation*}\big \{ {\textbf{x}} \in \mathbb{R}^d_+ \,:\, |{\textbf{x}}|_\infty > 1, \, x_i > \epsilon |{\textbf{x}}|_\infty \text{ for } i \in \beta, \, x_i \leq \epsilon |{\textbf{x}}|_\infty \text{ for } i \notin \beta \big\} \, .\end{equation*}

Subsequently, [Reference Goix, Sabourin and ClémenÇon18] introduced the notion of $\epsilon$ -thickened rectangles:

\begin{equation*}\big \{ {\textbf{x}} \in \mathbb{R}^d_+ \,:\, |{\textbf{x}}|_\infty > 1, \, x_i > \epsilon \text{ for } i \in \beta, \, x_i \leq \epsilon \text{ for } i \notin \beta \big\} \, .\end{equation*}

[Reference Chiapino and Sabourin5] relaxed the condition on $\beta^c$ and defined the rectangles

\begin{equation*}\big \{ {\textbf{x}} \in \mathbb{R}^d_+ \,:\, x_i > 1 \text{ for } i \in \beta \big\}\, ,\end{equation*}

to focus on groups of variables that may be large together. Similarly to [Reference Chiapino and Sabourin5], we relax the condition on $\beta$ in the definition of the $C_\beta$ , which leads to the study of the subsets

\begin{equation*}\{{\textbf{x}} \in \mathbb S^{d-1}_+ \,:\, x_i = 0 \text{ for } i \notin \beta\} = \mathbb S ^{d-1}_+ \cap \mathrm{Vect}({\textbf{e}}_j, j \in \beta).\end{equation*}

We gather in the following proposition some results regarding the behavior of $\pi({\textbf{X}}/t)$ and ${\textbf{Z}}$ on the subsets $C_\beta$ and $\mathbb S^{d-1}_+ \cap \mathrm{Vect}({\textbf{e}}_j, j \in \beta)$ . In all of what follows, we write $\Theta_{\beta, \, j}$ (resp. $\Theta_{\beta, \, j, \, +}$ ) for $\sum_{k \in \beta} (\Theta_k - \Theta_j)$ (resp. $\sum_{k \in \beta} (\Theta_k - \Theta_j)_+$ ).

Regarding the behavior of $\pi({\textbf{X}}/t)$ , the convergence (2.3) holds for any pair of Borel sets $A \times B \in (1, \infty) \times \mathbb S^{d-1}_+$ such that ${\mathbb{P}}(Y {\boldsymbol{\Theta}} \in \partial \pi^{-1}(B)) = 0$ , where $\partial \pi^{-1}(B)$ denotes the boundary of the set $\pi^{-1}(B)$ . The first part of Proposition 2 states that the subsets $C_\beta$ and $\mathbb S^{d-1}_+ \cap \mathrm{Vect}({\textbf{e}}_j, j \in \beta)$ satisfy this condition. This implies in particular that the sparsity of ${\textbf{Z}}$ can be studied through the projected vector $\pi({\textbf{X}}/t)$ . This will be illustrated in Section 4.

The second part of Proposition 2 provides some interesting relations between the sparsity of ${\textbf{Z}}$ and that of ${\boldsymbol{\Theta}}$ . If we consider $\beta = \{1,\ldots,d\}$ , then we obtain the probability that all coordinates are positive, which has already been computed in (2.7). It is equal to $G_{\textbf{Z}}({\textbf{0}}) = 1 - {\mathbb{E}} \left[ \max_{1\leq j \leq d } (1-d\Theta_j)^\alpha \right]$ . Another special case of Equations (3.4) and (3.5) is the one where $\beta$ corresponds to a single coordinate $\{j_0\}$ . In this case, since ${\textbf{Z}}$ belongs to the simplex, the two probabilities ${\mathbb{P}}({\textbf{Z}}_{\beta^c} = 0)$ and ${\mathbb{P}}({\textbf{Z}} \in C_\beta)$ are equal. Their common value corresponds to the probability that ${\textbf{Z}}$ concentrates on the $j_0$ th axis, which is equal to

(3.6) \begin{equation} {\mathbb{P}}( Z_{j_0} = 1) = {\mathbb{E}} \Big[ \min_{j \neq j_0} (\Theta_{j_0} - \Theta_j)_+^\alpha \Big] \, .\end{equation}

Then Equation (3.6) can be developed in the following way:

\begin{align*}{\mathbb{P}}( Z_{j_0} = 1)& = {\mathbb{E}} \Big[ \min_{j \neq j_0} (\Theta_{j_0} - \Theta_j)_+^\alpha {1}_{\{\Theta_{j_0} = 1\}} \Big]+ {\mathbb{E}} \Big[ \min_{j \neq j_0} (\Theta_{j_0} - \Theta_j)_+^\alpha {1}_{\{\Theta_{j_0} < 1\}} \Big]\\& = {\mathbb{P}}(\Theta_{j_0} = 1) + {\mathbb{E}} \Big[ \min_{j \neq j_0} (\Theta_{j_0} - \Theta_j)_+^\alpha {1}_{\Theta_{j_0} < 1} \Big] \geq {\mathbb{P}}(\Theta_{j_0} = 1) \, .\end{align*}

This shows that the vector ${\textbf{Z}}$ is more likely to be sparse than the spectral vector ${\boldsymbol{\Theta}}$ .

Remark 5. Following Equation (3.5), we write

(3.7) \begin{align} {\mathbb{P}}({\textbf{Z}}_{\beta^c} = 0) &\geq {\mathbb{E}}\Big[ \min_{j \in \beta^c} \Theta_{\{1,\ldots,d\}, \, j, \, +}^\alpha {1}_{\{ {\boldsymbol{\Theta}}_{\beta^c} = 0\} } \Big]\nonumber\\ &= {\mathbb{E}}\Bigg[ \bigg(\sum_{k \leq d} \Theta_k \bigg)^\alpha {1}_{\{ {\boldsymbol{\Theta}}_{\beta^c} = 0\} } \Bigg] = {\mathbb{P}}({\boldsymbol{\Theta}}_{\beta^c} = 0)\, . \end{align}

This can also be seen as a direct consequence of Property P2; see Section 2.1. This property also gives

(3.8) \begin{equation}{\mathbb{P}}({\textbf{Z}}_\beta > 0) \leq {\mathbb{P}}({\boldsymbol{\Theta}}_\beta > 0)\, , \end{equation}

an inequality which will be useful in some proofs.

3.2. Maximal directions

We focus in this section on the subsets $C_\beta$ . A positive value for ${\mathbb{P}}( {\boldsymbol{\Theta}} \in C_\beta)$ implies that the marginals $X_j$ for $j \in \beta$ take simultaneously large values while the ones in $\beta^c$ do not [Reference Chautru4, Reference Simpson, Wadsworth and Tawn34, Reference Goix, Sabourin and ClémenÇon18]. Our aim is to use Proposition 2 to compare the nullity or not of the probabilities ${\mathbb{P}}({\boldsymbol{\Theta}} \in C_\beta)$ and ${\mathbb{P}}({\textbf{Z}} \in C_\beta)$ . To this end, it is relevant to focus on the largest group of variables $\beta \in {{\mathcal{P}}}_d^*$ such that ${\mathbb{P}}({\boldsymbol{\Theta}} \in C_\beta) > 0$ . This motivates the notion of maximal direction.

Definition 2. (Maximal direction.) Let $\beta \in {{\mathcal{P}}}_d^*$ . We say that a direction $\beta$ is maximal for ${\boldsymbol{\Theta}}$ if

\begin{equation*} {\mathbb{P}}({\boldsymbol{\Theta}} \in C_\beta) > 0 \quad \text{and} \quad {\mathbb{P}}({\boldsymbol{\Theta}} \in C_{\beta'}) =0\, , \text{ for all } \beta' \supsetneq \beta\, . \end{equation*}

We define maximal directions for ${\textbf{Z}}$ similarly.

The notion of maximal directions is justified by the following theorem.

Theorem 2 implies that we do not lose any information on the extremal directions of $\beta$ by studying ${\textbf{Z}}$ instead of ${\boldsymbol{\Theta}}$ . But it is possible that the distribution of ${\textbf{Z}}$ puts some mass on a subset $C_\beta$ while that of ${\boldsymbol{\Theta}}$ does not. In such a case, the associated direction $\beta$ is necessarily non-maximal.

Example 3 shows there may exist $\beta \in {{\mathcal{P}}}_d^*$ such that ${\mathbb{P}}({\textbf{Z}} \in C_\beta) > 0$ and ${\mathbb{P}}({\boldsymbol{\Theta}} \in C_\beta) = 0$ . In this case, Theorem 2 states that the direction $\beta$ is not maximal for ${\textbf{Z}}$ since it is not maximal for ${\boldsymbol{\Theta}}$ . Following Remark 6, we consider a maximal direction $\gamma$ of ${\textbf{Z}}$ such that $\beta \subset \gamma$ . Then Theorem 2 states that ${\mathbb{P}}({\boldsymbol{\Theta}} \in C_\gamma) > 0$ . This means that even if the direction $\beta$ does not itself gather coordinates on which extreme values simultaneously occur, there exists a superset of $\beta$ which actually contains extremes. Thus, $\beta$ still gives information on the study of large events.

A natural procedure to capture the extremal directions of ${\textbf{X}}$ is then the following one. Based on the Euclidean projection $\pi$ we identify the subsets $C_\beta$ on which the distribution of ${\textbf{Z}}$ places mass. Hopefully, the selected subsets are low-dimensional. Among these subsets we select the maximal ones which also correspond to the maximal direction for the spectral vector ${\boldsymbol{\Theta}}$ .

What happens on non-maximal directions? While the study of maximal directions is the same for ${\textbf{Z}}$ and ${\boldsymbol{\Theta}}$ , we develop here some ideas which highlight the use of ${\textbf{Z}}$ for non-maximal directions. We consider a direction $\beta \in {{\mathcal{P}}}^*_d$ and assume that the associated subset $C_\beta$ satisfies ${\mathbb{P}}({\textbf{Z}} \in C_\beta) > 0$ and ${\mathbb{P}}({\boldsymbol{\Theta}} \in C_\beta) = 0$ . Then the direction $\beta$ is necessary non-maximal for ${\textbf{Z}}$ and satisfies the following inequalities:

\begin{equation*}0 < {\mathbb{P}}({\textbf{Z}} \in C_\beta)= {\mathbb{P}}( {\textbf{Z}}_\beta > 0, {\textbf{Z}}_{\beta^c} = 0)\leq {\mathbb{P}}( {\textbf{Z}}_{\beta^c} = 0)= {\mathbb{P}}( \pi(Y {\boldsymbol{\Theta}})_{\beta^c} = 0)\, .\end{equation*}

Following Equation (C.20) we obtain that

(3.9) \begin{equation}0 < {\mathbb{P}}( \pi(Y {\boldsymbol{\Theta}})_{\beta^c} = 0)= {\mathbb{P}} \big( |\beta|^{-1} |{\boldsymbol{\Theta}}_\beta| \geq \max_{i \in \beta^c} \Theta_i + Y^{-1} \big)\, .\end{equation}

This implies that with positive probability $|\beta|^{-1} \sum_{k \in \beta} \Theta_k \geq \Theta_i$ for all $i \in \beta^c$ . If we consider for instance $\beta = \{j\}$ , then we obtain that with positive probability $\Theta_j \geq \Theta_i$ for all $i \neq j$ . More generally, regarding the vector ${\textbf{X}}$ , Equation (3.9) yields

\begin{equation*}0 < {\mathbb{P}} \big( |\beta|^{-1} |{\boldsymbol{\Theta}}_\beta| \geq \max_{i \in \beta^c} \Theta_i + Y^{-1} \big)= \lim\limits_{t \to \infty}{\mathbb{P}} \big( |\beta|^{-1} |{\textbf{X}}_\beta| \geq \max_{i \in \beta^c} X_i + t \mid |{\textbf{X}}| > t \big)\, .\end{equation*}

This means that the extreme values of ${\textbf{X}}_\beta$ are likely to be larger than the extreme values of ${\textbf{X}}_{\beta^c}$ . This does not contradict the fact that ${\mathbb{P}}({\boldsymbol{\Theta}} \in C_\beta) = 0$ , which only implies that it is unlikely that one will simultaneously observe large values in the direction $\beta$ and small values in the direction $\beta^c$ .

Hence, if we detect a maximal direction $\gamma$ , we first infer that it is likely that the directions in $\gamma$ are large together while those in $\gamma^c$ take small values. The marginals in $\gamma$ form a cluster of extremal directions for which the relative importance of each direction can be studied via the identification of non-maximal directions $\beta \subsetneq \gamma$ such that ${\mathbb{P}}( {\textbf{Z}} \in C_\beta) > 0$ . Indeed, if we detect such a subset it means that in the cluster $\gamma$ the directions in $\beta$ are likely to be larger than the ones in $\gamma \setminus \beta$ . A deeper interpretation of non-maximal directions is deferred to future work.

We first consider the direction $\{1\}$ and compute the probability that ${\textbf{Z}}$ belongs to the subset $C_{ \{1\} }$ . Following Equation (C.3), we obtain that

\begin{equation*} {\mathbb{P}}\big( {\textbf{Z}} \in C_{ \{1\} } \big) ={\mathbb{P}}\Big( \min _{j \geq 2} \, (Y \Theta_1 - Y \Theta_j) \geq 1 \Big) = {\mathbb{P}}\Big( Y \Theta_1 \geq \max_{j \geq 2} \, Y \Theta_j +1 \Big)\, . \end{equation*}

Then we use the relation ${\boldsymbol{\Theta}} = {\textbf{a}}$ a.s., which implies that

\begin{equation*} {\mathbb{P}}\big( {\textbf{Z}} \in C_{ \{1\} } \big) = {\mathbb{P}}\big( Y a_1\geq Y a_2+ 1 \big) = {\mathbb{P}}\big( Y \geq (a_1-a_2)^{-1} \big) = (a_1-a_2)^\alpha\, . \end{equation*}

Hence, the probability that ${\textbf{Z}}$ belongs to the first axis depends on the difference between the first and the second coordinate of ${\textbf{a}}$ .

More generally, for $1 \leq r \leq d-1$ , Equation (C.4) implies that ${\textbf{Z}}$ belongs to the subset $C_{ \{1, \ldots, r\} }$ if and only if

\begin{equation*} \max_{1 \leq i \leq r}\, \sum_{j=1}^r (Y \Theta_j - Y \Theta_i) < 1\, \quad \text{and} \quad \min_{r+1 \leq i \leq d}\, \sum_{j=1}^r (Y\Theta_j - Y\Theta_i) \geq 1 \, . \end{equation*}

Thus, the probability that ${\textbf{Z}}$ belongs to the subset $C_{\{1, \ldots, r\}}$ is equal to

\begin{align*} {\mathbb{P}}\big( {\textbf{Z}} \in C_{ \{1, \ldots, r\} } \big) &={\mathbb{P}}\bigg( \sum_{j=1}^r (Y a_j - Y a_r) < 1, \, \sum_{j=1}^r (Y\Theta_j - Y\Theta_{r+1}) \geq 1\bigg)\\ &= {\mathbb{P}}\big( (\tilde a_r - r a_{r+1})^{-1} \leq Y < (\tilde a_r - a_r)^{-1} \big) \quad \text{where } \tilde a_r = a_1 + \ldots + a_r\\ &= (\tilde a_r - r a_{r+1})^\alpha - (\tilde a_r - r a_r)^\alpha\, . \end{align*}

If we take $\alpha = 1$ for the sake of simplicity, then we obtain ${\mathbb{P}} ( {\textbf{Z}} \in C_{ \{1, \ldots, r\} } ) = r (a_r - a_{r+1})$ , and thus the probability that ${\textbf{Z}}$ belongs to the subset $C_{\{1, \ldots, r\}}$ depends only on the distance between $a_r$ and $a_{r+1}$ .

This example emphasizes the use of the vector ${\textbf{Z}}$ on non-maximal directions. It highlights the relative importance of a coordinate in relation to the extreme values of a group of directions this coordinate belongs to.

4.1. The framework

We consider an independent and identically distributed (i.i.d.) sequence of regularly varying random vectors ${\textbf{X}}_1, \ldots, {\textbf{X}}_n$ with generic distribution ${\textbf{X}} \in \mathrm{RV}(\alpha, {\boldsymbol{\Theta}})$ . We set ${\textbf{Z}} = \pi(Y {\boldsymbol{\Theta}})$ , where Y follows a Pareto( $\alpha$ ) distribution independent of ${\boldsymbol{\Theta}}$ . Our aim is to capture the directions $\beta \in {{\mathcal{P}}}_d^*$ such that ${\mathbb{P}}({\textbf{Z}} \in C_\beta ) >0$ . Thanks to Proposition 2, the latter probability is defined through the limit

(4.1) \begin{equation} {\mathbb{P}}({\textbf{Z}} \in C_\beta )= \lim\limits_{t \to \infty} {\mathbb{P}}(\pi({\textbf{X}} / t) \in C_\beta \mid |{\textbf{X}}| > t)= \lim\limits_{t \to \infty} \frac{{\mathbb{P}}(\pi({\textbf{X}} / t) \in C_\beta, \, |{\textbf{X}}| > t)}{{\mathbb{P}}( |{\textbf{X}}| > t)} \, .\end{equation}

The goal is then to approximate this probability with the sample ${\textbf{X}}_1, \ldots, {\textbf{X}}_n$ . We define the quantity

(4.2) \begin{equation} T_\beta(t) = \frac{ \sum_{j = 1}^n {1} \{ \pi({\textbf{X}}_j / t) \in C_\beta, |{\textbf{X}}_j| > t \} }{ \sum_{j = 1}^n {1} \{|{\textbf{X}}_j| > t \} }\, , \quad t > 0\, , \quad \beta \in {{\mathcal{P}}}_d^*\, ,\end{equation}

which corresponds to the proportion of data ${\textbf{X}}_j$ whose projected vector $\pi({\textbf{X}}_j /t)$ belongs to $C_\beta$ among the data whose $\ell^1$ -norm is above t. Intuitively, the larger the variable $T_\beta(t)$ , the more likely the direction $\beta$ gathers extreme values. The law of large numbers then implies the following approximation:

\begin{equation*}T_\beta(t)\approx\frac{{\mathbb{P}}(\pi({\textbf{X}} / t) \in C_\beta, \, |{\textbf{X}}| > t)}{{\mathbb{P}}( |{\textbf{X}}| > t)}\approx{\mathbb{P}}({\textbf{Z}} \in C_\beta )\, ,\end{equation*}

where the first approximation holds for n large and the second one for t large. This approximation allows one to classify the directions $\beta$ depending on the nullity or not of the associated quantity $T_\beta(t)$ . Actually, once t is fixed, we can get rid of the denominator in (4.2) and focus only on $\sum_{j = 1}^n {1} \{ \pi({\textbf{X}}_j / t) \in C_\beta, |{\textbf{X}}_j| > t \}$ . The selection of the largest vectors ${\textbf{X}}_j$ whose norm is above t then boils down to keeping only a proportion, say k, of vectors. It is customary in EVT to choose a level k which satisfies $k \to \infty$ and $k/n \to 0$ when $n \to \infty$ .

Remark 7. (The approach proposed by [Reference Goix, Sabourin and ClémenÇon18].) In order to detect anomalies among multivariate extremes, [Reference Goix, Sabourin and ClémenÇon18] propose a similar approach with the $\ell^\infty$ -norm based on the $\epsilon$ -thickened rectangles

\begin{equation*} R^\epsilon_\beta = \{ {\textbf{x}} \in \mathbb{R}^d_+ \,:\, |{\textbf{x}}|_\infty > 1, \,x_j > \epsilon \text{ for all } j \in \beta, \, x_j \leq \epsilon \text{ for all } j \in \beta^c \}\, , \quad \beta \in {{\mathcal{P}}}_d^*\, . \end{equation*}

Starting from the sample ${\textbf{X}}_1, \ldots, {\textbf{X}}_n$ with generic random vector ${\textbf{X}} = (X^1, \ldots, X^d)$ with marginal distribution $F_1, \ldots, F_d$ , the authors define the vectors ${\textbf{V}}_i = (1 / (1-\hat {F_j} (X_i^j)))_{j = 1, \ldots, d}$ for $i = 1, \ldots, n$ , where

\begin{equation*}\hat {F_j} \,:\, x \mapsto \frac{1}{n} \sum_{i=1}^n {1}_{X_i^j < x}\end{equation*}

is the empirical counterpart of $F_j$ . This rank transformation provides standardized marginals to the vectors ${\textbf{V}}_i$ . Denoting by $\tilde {\boldsymbol{\Theta}}_\infty$ the nonstandard spectral vector with respect to the $\ell^\infty$ -norm and by $C_{\beta, \infty} = \big \{ {\textbf{x}} \in \mathbb S^d_{+, \infty}, \, x_i > 0 \text{ for } i \in \beta, \, x_i = 0 \text{ for } i \notin \beta \big\}$ the associated subsets, [Reference Goix, Sabourin and ClémenÇon18] use the approximation

\begin{equation*} T_\beta(k, \epsilon) = \frac{1}{k} \sum_{i=1}^n {1}_{{\textbf{V}}_i \in (n/k) R^\epsilon_\beta} \approx c \, {\mathbb{P}}( \tilde {\boldsymbol{\Theta}}_\infty \in C_{\beta, \infty})\, , \quad c > 0\, , \end{equation*}

for k large and $\epsilon$ close to zero (the ratio $n/k$ plays the role of the large threshold t; see [Reference De Haan and Ferreira20]). The authors propose an algorithm called DAMEX whose goal is to identify the subsets $C_{\beta, \infty}$ such that ${\mathbb{P}}( \tilde {\boldsymbol{\Theta}}_\infty \in C_{\beta, \infty}) > 0$ .

Remark 8. (On the choice of the norm.) After some calculations we observe that if the spectral vectors $\tilde {\boldsymbol{\Theta}}$ and $\tilde {\boldsymbol{\Theta}}_\infty$ correspond to nonstandard regular variation (with $\alpha = 1$ ), then they satisfy the relation

(4.3) \begin{equation}{\mathbb{P}}(\tilde {\boldsymbol{\Theta}} \in B) = \frac{ {\mathbb{E}} \big[|\tilde {\boldsymbol{\Theta}}_\infty| {1}_{ \{\tilde {\boldsymbol{\Theta}}_\infty / |\tilde {\boldsymbol{\Theta}}_\infty| \in B\} } \big] }{ {\mathbb{E}}[|\tilde {\boldsymbol{\Theta}}_\infty| ] }\, \end{equation}

for all $B \in \mathbb S^{d-1}_+$ . Since $\tilde {\boldsymbol{\Theta}}_\infty / |\tilde {\boldsymbol{\Theta}}_\infty| \in C_\beta$ if and only if $\tilde {\boldsymbol{\Theta}}_\infty \in C_{\beta, \infty}$ , we obtain the equivalence

\begin{equation*} {\mathbb{P}}(\tilde {\boldsymbol{\Theta}} \in C_\beta) > 0 \quad \text{if and only if} \quad {\mathbb{P}}(\tilde {\boldsymbol{\Theta}}_\infty \in C_{\beta, \infty}) > 0\, . \end{equation*}

Hence, the directions in which extremes gather are the same regardless the choice of the norm. This means that after a standardization of the marginals we can compare the performance of our method with that of [Reference Goix, Sabourin and ClémenÇon18]. This is what we do in the first numerical example below.

Note that if ${\textbf{X}}$ is regularly varying with tail index 1, then ${\textbf{X}}^q$ is regularly varying with tail index $1/q$ and Equation (4.3) implies that the corresponding spectral measures concentrate on the same subsets $C_\beta$ (see also Remark 11).

Taking this hyperparameter p into account, we are now able to introduce the algorithm used to study the dependence structure of sparsely regularly varying random vectors.

Algorithm 1:

Extremal dependence structure of sparsely regularly varying random vectors.

4.2. Experimental results

In this section we consider two different cases of numerical data for which we apply Algorithm 1. For each case we generate data sets of size $n \in \{10^4, \ 5 \cdot 10^4, \ 10^5\}$ , we compute the quantities $T_\beta(t)$ , and we repeat this procedure over $N =100$ simulations. Regarding the outcome ${\mathcal{C}} = \{\beta, \, T_\beta(t) > 0 \}$ of our procedure, two different types of errors could arise. The first one corresponds to the occurrence of a direction $\beta$ that theoretically should not appear. Such an error will be called an error of type 1. The second type of error corresponds to the absence of a direction $\beta$ that theoretically should appear. Such an error will be called an error of type 2. The results correspond to the average number of each error among the N simulations. The code can be found at https://github.com/meyernicolas/projection_extremes.

The purpose of the experiments is to study the procedure given in Algorithm 1 and see how it manages to detect the sparsity in the extremes. We also analyze the influence of the tail index $\alpha$ by choosing different values for this parameter. This is done by considering the random vector ${\textbf{X}}^\alpha =(X_1^\alpha, \ldots, X_d^\alpha)$ whose tail index is $1/\alpha$ for ${\textbf{X}} \in \mathrm{RV}(1, {\boldsymbol{\Theta}})$ .

Remark 11. Assume that $\pi({\textbf{X}}/t) \in C_\beta$ . This implies that for all $j \in \beta$ we have $X_j > \max_{i \in \beta^c} X_i$ and $|{\textbf{X}}_\beta| - \max_{i \in \beta^c} X_i > t$ ; see (C.4). Hence for $k \in \beta$ and $q \geq 1$ we obtain

\begin{align*} |{\textbf{X}}^q_\beta| - \max_{i \in \beta^c} X_i^q &= X_k^q - \big(\max_{i \in \beta^c} X_i\big)^q + |{\textbf{X}}^q_{\beta \setminus \{k\}}|\\ &= \big( X_k - \big(\max_{i \in \beta^c} X_i\big) \big) \sum_{l=0}^q X_k^l \big(\max_{i \in \beta^c} X_i\big)^{q-l-1} + |{\textbf{X}}^q_{\beta \setminus \{k\}}|\\ &\geq X_k - \big(\max_{i \in \beta^c} X_i\big) + |{\textbf{X}}_{\beta \setminus \{k\}}| = |{\textbf{X}}_\beta| - \big(\max_{i \in \beta^c} X_i\big) \geq t \, , \end{align*}

as soon as we assume that all marginals satisfy $X_j > 1$ . The relation $|{\textbf{X}}^q_\beta| - \max_{i \in \beta^c} X_i^q$ implies that $\pi({\textbf{X}}^q/t)_{\beta^c} = 0$ ; see (C.3). In other words, it means that $\pi({\textbf{X}}^q/t)$ belongs to $C_\gamma$ for $\gamma \subset \beta$ .

In particular, if the spectral measure of ${\textbf{X}}$ only concentrates on the axis, these directions can be more easily detected through the study of $\pi({\textbf{X}}^q/t)$ , $q \geq 1$ , than through that of $\pi({\textbf{X}}/t)$ .

Asymptotic independence. Let ${\textbf{N}}_1, \ldots, {\textbf{N}}_n$ be an i.i.d. sequence of random vectors in $\mathbb{R}^{40}$ with generic random vector ${\textbf{N}}$ whose distribution is a multivariate Gaussian distribution with all marginals following a univariate standard Gaussian distribution and the correlations less than 1: ${\mathbb{E}}[N^i N^j] < 1$ for all $1 \leq i \neq j \leq d$ . We transform the marginals with a rank transform which consists in considering the vectors ${\textbf{X}}_1, \ldots, {\textbf{X}}_n$ such that the marginals $X^j_i$ of ${\textbf{X}}_i = (X_i^1, \ldots, X_i^d)$ are defined as

\begin{equation*}X_i^j = \frac{1}{1 - \hat{F_j}(N^j_i)}\, , \quad 1 \leq j \leq d\, ,\end{equation*}

where $\hat{F_j}$ is the empirical version of the cumulative distribution function of $N_j \sim {{\mathcal{N}}}(0,1)$ . This provides a sample of regularly varying random vectors ${\textbf{X}}_1, \ldots, {\textbf{X}}_n$ , and the assumption on the correlation leads to asymptotic independence, i.e. ${\mathbb{P}}({\boldsymbol{\Theta}} \in C_\beta) = {\mathbb{P}}({\textbf{Z}} \in C_\beta) = 1/d$ for all $\beta$ such that $|\beta| = 1$ (see [Reference Sibuya33]). The aim of our procedure is to recover these 40 directions from among the $2^{40} -1 \approx 10^{12}$ directions.

Regarding the multivariate Gaussian random vectors ${\textbf{N}}_1, \ldots, {\textbf{N}}_n$ , the simulation of these vectors depends only on their covariance matrix. We proceed as follows. We generate a matrix ${\boldsymbol \Sigma}'$ with entries $\sigma^{\prime}_{i,j}$ following independent uniform distributions on $(-1,1)$ . Then we define the matrix ${\boldsymbol \Sigma}$ as

\begin{equation*}{\boldsymbol \Sigma} = \mathrm{Diag}\Big({\sigma^{\prime}}_{1,1}^{-1/2}, \ldots, {\sigma^{\prime}}_{d,d}^{-1/2}\Big) \cdot {\boldsymbol \Sigma}'^T \cdot {\boldsymbol \Sigma}' \cdot \mathrm{Diag}\Big({\sigma^{\prime}}_{1,1}^{-1/2}, \ldots, {\sigma^{\prime}}_{d,d}^{-1/2}\Big)\, ,\end{equation*}

where $\mathrm{Diag}(\sigma_{1,1}^{-1/2}, \ldots, \sigma_{d,d}^{-1/2})$ denotes the diagonal matrix of ${{\mathcal{M}}}_d(\mathbb{R})$ whose diagonal is given by the vector $({\sigma^{\prime}}_{1,1}^{-1/2}, \ldots, {\sigma^{\prime}}_{d,d}^{-1/2})$ . This provides a covariance matrix with diagonal entries equal to 1 and off-diagonal entries less than 1. A given matrix ${\boldsymbol \Sigma}$ then provides a dependence structure for ${\textbf{N}}$ and thus for ${\textbf{X}}$ . We generate $N_{\text{model}} = 20$ different matrices ${\boldsymbol \Sigma}$ , and for each of these dependence structures we generate $N=100$ sample ${\textbf{N}}_1, \ldots, {\textbf{N}}_n$ . We summarize in Table 1 and in Table 2 the two types of errors averaged among the $N \cdot N_{\text{model}} = 2000$ simulations.

Table 1.

Average number of errors of type 1 in an asymptotic independence case ( $d=40$ ).

Table 2.

Average number of errors of type 2 in an asymptotic independence case ( $d=40$ ).

In this case, the standard spectral vector ${\boldsymbol{\Theta}}$ and the non-standard one $\tilde {\boldsymbol{\Theta}}$ coincide. Hence it is possible to compare ${\boldsymbol{\Theta}}$ and $\tilde {\boldsymbol{\Theta}}_\infty$ (see Remark 8), which is done by computing the quantities $T_\beta(k, \epsilon)$ as well as the two types of errors for the DAMEX algorithm. We only study the effect of $\alpha$ for our approach, since the DAMEX algorithm is not sensitive to marginals.

For the Euclidean projection with $\alpha = 1$ we observe that our algorithm manages to capture almost all $d=40$ directions regardless the value of n, and the number of errors of type 2 decreases when n increases. On the other hand, our algorithm still captures some extra directions, especially for $n = 10^4$ and $n = 5 \cdot 10^4$ . This may be a consequence of the choice of p in Remark 9, which is probably too high and for which a deeper study should be conducted. The number of errors of type 1 is then much lower for $n =10^5$ . We observe that for $\alpha=1/2$ we obtain better results, while the number of errors is higher for $\alpha =2$ . Since the extremal directions are in this case only one-dimensional, this numerically confirms the observations of Remark 11.

Regarding the DAMEX algorithm, a large $\epsilon$ theoretically leads to more mass assigned on the axis. This explains why in our simulations choosing a large $\epsilon$ reduces the number of errors of type 2. With $\epsilon = 0.5$ the algorithm manages to capture all $d=40$ axes; however, the number of errors of type 1 is quite large, regardless of the choice of n. Hence it seems that our procedure leads to the best compromise between the two types of errors.

A dependent case. We now consider a dependent case where extremes occur on lower-dimensional directions. In order to include dependence we recall that a vector ${\textbf{P}}(k) =(P_1, P_1 + P_2, \ldots, P_1 + P_k) \in \mathbb{R}^k_+$ with $P_j$ following Pareto( $\alpha_j$ ), $\alpha_1 < \alpha_j$ for all $ 2 \leq j \leq k$ , is regularly varying with tail index $\alpha_1$ and spectral vector ${\boldsymbol{\Theta}} = (1/k,\ldots, 1/k)$ . For our simulations we consider $s_1=10$ independent copies ${\textbf{P}}_1, \ldots, {\textbf{P}}_{s_1}$ of ${\textbf{P}}(2) \in \mathbb{R}^2_+$ with $\alpha_1 = 1$ and $\alpha_2 = 2$ and $s_2=10$ independent copies ${\textbf{R}}_1, \ldots, {\textbf{R}}_{s_2}$ of ${\textbf{P}}(3) \in \mathbb{R}^3_+$ with $\alpha_1 = 1$ and $\alpha_2 = \alpha_3 = 2$ . We aggregate these vectors and form a vector ${\textbf{X}}$ in $\mathbb{R}^{50}_+$ which is then regularly varying with a discrete spectral measure placing mass on the points $({\textbf{e}}_j + {\textbf{e}}_{j+1})/2$ for $j = 1, 3, \ldots, 17, 19$ and on the points $({\textbf{e}}_j + {\textbf{e}}_{j+1} + {\textbf{e}}_{j+2}) / 3$ for $j = 21, 24, \ldots, 45, 48$ . As discussed after Proposition 1, in this case the angular vector ${\textbf{Z}}$ and the spectral vector ${\boldsymbol{\Theta}}$ are equal a.s. Our aim is to recover the $s_1 = 10$ two-dimensional directions $({\textbf{e}}_j + {\textbf{e}}_{j+1})/2$ for $j = 1, 3, \ldots, 17, 19$ and also the $s_2 =10$ three-dimensional directions $({\textbf{e}}_j + {\textbf{e}}_{j+1} + {\textbf{e}}_{j+2}) / 3$ for $j = 21, 24, \ldots, 45, 48$ , based on a sample of i.i.d. random vectors ${\textbf{X}}_1, \ldots, {\textbf{X}}_n$ with the same distribution as ${\textbf{X}}$ . Hence we would like to recover $s = s_1 + s_2 = 20$ directions from among the total of $2^{50}-1 \approx 10^{15}$ directions.

As for asymptotic independence, we remark that the number of errors decreases when n increases for almost all cases. For $\alpha = 1$ and $\alpha = 1/2$ our algorithm not only is able to detect all $s = 20$ directions on which the distribution of ${\textbf{Z}}$ puts mass, but also does not identify any extra directions. These results are all the more accurate since the identification of the $s = 20$ directions is done among a very large number of directions, in this case $2^{50} - 1 \approx 10^{15}$ . However, the fact that we obtain better results for the errors of type 1 in the case $\alpha = 1/2$ than in the case $\alpha = 1$ cannot be explained by Remark 11. We defer this question to future work.

4.3. Sparse regular variation and non-maximal directions

In this section we illustrate some interpretations of the vector ${\textbf{Z}}$ in relation to extremal directions in non-maximal directions (see the discussion in Section 3.2). We consider a vector ${\textbf{a}} \in \mathbb S^{r-1}_+$ as in Example 4 and a Pareto $(\alpha)$ -distributed random variable P, and define $P {\textbf{a}} = (a_1 P, \ldots, a_r P) \in \mathrm{RV}(\alpha, {\textbf{a}})$ . Then, combining this device with that of the dependent case, we consider ${\textbf{P}} = (a_1 P, a_2 P + P_2, \dots, a_r P + P_r) \in \mathrm{RV}(\alpha, {\textbf{a}})$ , where $P_2, \ldots, P_r$ are i.i.d. random variables following a Pareto distribution with parameter $\alpha' > \alpha$ . Hence the degenerate spectral vector ${\boldsymbol{\Theta}} = {\textbf{a}}$ places mass only in the direction ${ \{1, \ldots, r\} }$ , which is thus maximal, while the vector ${\textbf{Z}}$ places mass in all non-maximal directions $\{1\}, \{1, 2\}, \{1, 2, 3\}, \ldots, \{1, \ldots, d-1\}$ ; see Example. In our simulations, we choose $\alpha =1$ , $\alpha' = 2$ , and $r=3$ , and we consider a vector ${\textbf{a}} = (7,6,4)/|(7,6,4)|$ . We then aggregate $s=20$ i.i.d. copies ${\textbf{P}}_k$ of the vector ${\textbf{P}}$ and obtain ${\textbf{X}} = ({\textbf{P}}_1, \ldots, {\textbf{P}}_s) \in \mathrm{RV}(1, {\boldsymbol{\Theta}})$ with ${\boldsymbol{\Theta}}$ placing mass on the three-dimensional maximal directions ${ \{j, j+1, j+2\} }$ for $j \in J = \{1, 4, 7, \ldots, 58\}$ , while the angular vector ${\textbf{Z}}$ places mass on the aforementioned maximal directions but also on $s=20$ two-dimensional and $s=20$ one-dimensional directions.

Table 3.

Average number of errors of type 1 and 2 in a dependent case ( $d = 50$ ).

The aim of the simulations is to see to what extent our procedure manages to recover the 40 non-maximal directions aforementioned. The columns of Table 4 give the average number of directions recovered by our algorithm, depending on their size. Recall that for each type of direction, the theoretical number of these directions that should appear is $s=20$ . Finally, the last column deals with the number of directions that theoretically should not appear. All the results are averaged across the $N=100$ simulations.

Table 4.

Average number of directions recovered by Algorithm 1 ( $d=60$ ).

For $n= 10^4$ we observe that the procedure manages to identify most of the one-dimensional directions, while the average number of two-dimensional directions is much smaller than the theoretical one. The same occurs for the maximal directions, for which we only manage to recover two-thirds of the theoretical ones. We also obtain a non-negligible number of extra directions which should not be identified. For $n = 5 \cdot 10^4$ , the three types of directions are quite well recovered by our algorithm, with once again very good results for the one-dimensional ones. The number of extra directions is still relatively high. For $n= 10^5$ , we keep a high level of accuracy in identifying all three types of directions, while the number of extra directions drastically decreases.

This example highlights the relevance of our approach not only to identifying clusters of directions that are simultaneously large, but also to studying the relative importance of the coordinates in a given cluster. This second aspect provides a deeper interpretation of ${\textbf{Z}}$ in terms of extremes.

C.1 Some results on the projection

We start with this section with three lemmas which gather some useful properties satisfied by the projection $\pi$ .

Proof of Lemma 1. The proof of this result relies on the relation $\pi_z({\textbf{v}}) = z \pi({\textbf{v}}/z)$ and on the characterization (A.2).

First we simplify the problem via the equivalences

\begin{align*} \forall \, 0 < z \leq z', \, \forall \, {\textbf{v}} \in \mathbb{R}^d_+, \, \pi_z( \pi_{z'} ({\textbf{v}}) ) &= \pi_z( {\textbf{v}} )\\ \iff &\forall \, 0 < z \leq z', \, \forall \, {\textbf{v}} \in \mathbb{R}^d_+, \, z \pi( z^{-1} \pi_{z'} ({\textbf{v}}) ) = z \pi( {\textbf{v}} / z)\\ \iff &\forall \, 0 < z \leq z', \, \forall \, {\textbf{v}} \in \mathbb{R}^d_+, \, \pi( z' z^{-1} \pi ({\textbf{v}} / z') ) = \pi( {\textbf{v}} / z)\\ \iff &\forall \, a \geq 1, \, \forall \, {\textbf{u}} \in \mathbb{R}^d_+, \, \pi (a \pi ({\textbf{u}})) = \pi( a {\textbf{u}})\, . \end{align*}

So we fix $a \geq 1$ and ${\textbf{u}} \in \mathbb{R}^d_+$ , and we prove this last equality by proving first that $\rho_{a \pi({\textbf{u}})} = \rho_{a {\textbf{u}}}$ and second that the positive coordinates of the two vectors $\pi(a \pi({\textbf{u}}))$ and $\pi(a {\textbf{u}})$ coincide.

STEP 1: We prove that $\rho_{a \pi({\textbf{u}})} = \rho_{a {\textbf{u}}}$ .

The characterization (A.2) entails that a coordinate j satisfies $\pi(a \pi({\textbf{u}}))_j > 0$ if and only if

(C.1) \begin{equation}a \pi({\textbf{u}})_j > \frac{1}{\sum_{k\leq d} {1}_{a \pi({\textbf{u}})_k \geq a \pi({\textbf{u}})_j} } \bigg( \sum_{k=1}^d a \pi({\textbf{u}})_k {1}_{a \pi({\textbf{u}})_k \geq a \pi({\textbf{u}})_j} -1 \bigg)\, . \end{equation}

Since $a \geq 1$ this assumption holds only if $\pi({\textbf{u}})_j = u - \lambda_{\textbf{u}} > 0$ . Hence, since $\pi$ preserves the order of the coordinates, we obtain that (C.1) is equivalent to

(C.2) \begin{equation}a (u_j -\lambda_{\textbf{u}}) > \frac{1}{\sum_{k\leq d} {1}_{a u_k \geq a u_j} } \bigg( \sum_{k=1}^d a (u_k - \lambda_{\textbf{u}}) {1}_{a u_k \geq a u_j} -1 \bigg)\, . \end{equation}

The terms with $\lambda_{\textbf{u}}$ vanish and (C.2) is then equivalent to $\pi(a{\textbf{u}})_j > 0$ . Hence the vectors $\pi( a \pi({\textbf{u}}) )$ and $\pi( a \pi({\textbf{u}}) )$ have the same positive coordinates, i.e. $\rho_{a \pi({\textbf{u}})} = \rho_{a {\textbf{u}}}$ .

STEP 2: We prove that $\pi( a \pi ({\textbf{u}}) ) = \pi(a {\textbf{u}})$ .

Let us denote by $\gamma$ the set of coordinates j such that $\pi( a \pi ({\textbf{u}}) )_j > 0$ . Step 1 ensures that this corresponds to the set of coordinates j such that $\pi(a {\textbf{u}})_j > 0$ . We prove that these two components coincide. Recall that for $k \in \gamma$ we have $\pi({\textbf{u}})_k = u_k - \lambda_{\textbf{u}} > 0$ . Then the result follows from the equalities

\begin{align*} \pi( a \pi ({\textbf{u}}) )_j = a \pi ({\textbf{u}})_j - \frac{|a \pi({\textbf{u}})_\gamma|-1}{|\gamma|} &= a (u_j - \lambda_{\textbf{u}}) - \frac{|a {\textbf{u}}_\gamma| - a\lambda_{\textbf{u}} |\gamma| -1}{|\gamma|}\\ &= a u_j - \frac{|a {\textbf{u}}_\gamma| -1}{|\gamma|} = \pi(a{\textbf{u}})_j\, , \end{align*}

for $j \in \gamma$ .

In the following lemma we compare the behavior of the vectors ${\textbf{u}}/|{\textbf{u}}|$ and $\pi({\textbf{u}}/t)$ on the sets $A_{\textbf{x}}$ .

Proof of Lemma 2.

1. 1. The assumption $\pi({\textbf{u}}/t) \in C_\gamma$ implies that $\lambda_{{\textbf{u}}/t} = (|{\textbf{u}}_\gamma|/t-1)/|\gamma|$ , so that for any $j \leq d$ we have

   \begin{equation*} \pi({\textbf{u}}/t)_j = \max ( u_j / t - \lambda_{{\textbf{u}}/t}, 0 ) \geq \frac{u_j}{t} - \frac{|{\textbf{u}}_\gamma|/t - 1}{|\gamma|} > \frac{u_j}{|{\textbf{u}}|} - \frac{|{\textbf{u}}|/t - 1}{|\gamma|} \geq \frac{u_j}{|{\textbf{u}}|} - \frac{\epsilon}{|\gamma|} \, , \end{equation*}
   where we used that $1<|{\textbf{u}}|/t \leq 1+\epsilon$ . Then the assumption ${\textbf{u}}/|{\textbf{u}}| \in A_{\textbf{x}}$ concludes the proof.

2. 2. If $\pi({\textbf{u}}/t) \in A_{{\textbf{x}}(1+\epsilon)}$ , then we have the inequality

   \begin{equation*} \qquad\qquad\qquad\qquad \frac{u_j}{|{\textbf{u}}|} = \frac{t}{|{\textbf{u}}|} \frac{u_j}{t} \geq \frac{1}{1+\epsilon} \pi({\textbf{u}}/t)_j \geq x_j\, , \quad 1 \leq j \leq d\, .\qquad\qquad\qquad\quad \end{equation*}

For $\gamma\in {{\mathcal{P}}}_d^*$ recall that the function $\phi_\gamma \,:\, \mathbb S^{d-1}_+ \to \mathbb{R}^d_+$ is defined by

\begin{equation*}\phi_\gamma({\textbf{u}})_j= \left\{\begin{array}{ll}&u_j + \frac{|{\textbf{u}}_{\gamma^c}|}{|\gamma|}\, , \quad j \in \gamma\, ,\\[8pt] &u_j + \frac{|{\textbf{u}}_{\gamma^c \setminus \{j\} }|}{|\gamma|+1} \, , \quad j \in \gamma^c\, .\end{array}\right.\end{equation*}

Also, we have defined the quantities $v_{\beta, \, i} = \sum_{j \in \beta} (v_j - v_i)$ and $v_{\beta, \, i, \, +} = \sum_{j \in \beta} (v_j - v_i)_+$ for ${\textbf{v}} \in \mathbb S^{d-1}_+$ , $\beta \in {{\mathcal{P}}}_d^*$ , and $1 \leq i \leq d$ .

Proof of Lemma 3.1.

1. 1. The characterization (A.2) ensures that $\pi({\textbf{v}})_i = 0$ if and only if

   \begin{equation*} v_i - \frac{1}{\sum_{k \leq d} 1_{v_k \geq v_i}} \bigg( \sum_{k\leq d} v_k{1}_{v_r \geq v_i} -1 \bigg) \leq 0\, , \end{equation*}
   which can be rephrased as
   \begin{equation*} 1 \leq \sum_{k=1}^d (v_k - v_i){1}_{v_k \geq v_i} \, . \end{equation*}
   This proves (C.3).

   For (C.4) the assumption $\pi({\textbf{v}}) \in C_\beta$ can be rephrased as follows:

   \begin{equation*} \forall i \in \beta, \, v_i = \pi({\textbf{v}})_i + \big(|{\textbf{v}}_\beta| - 1\big) / |\beta|\quad \text{and} \quad \forall i \in \beta^c, \, v_i \leq (|{\textbf{v}}_\beta| - 1\big) / |\beta| \, . \end{equation*}
   On the one hand, since $\pi({\textbf{v}})_i > 0$ for $i \in \beta$ , the first equality is equivalent to $\max_{i \in \beta} \sum_{j \in \beta} (v_j-v_i) < 1$ . On the other hand, the second equality is equivalent to $\min_{i \in \beta^c} \sum_{j \in \beta} (v_j-v_i) \geq 1$ . This proves (C.4).

2. 2. By definition of the projection $\pi$ the following equivalence holds:

   \begin{equation*} \left\{ \begin{array}{lll} &\pi(a {\textbf{u}})_j \geq x_j \, , \quad j \in \beta\, ,\\[7pt] &\pi(a {\textbf{u}})_j > 0 \, , \quad j \in \gamma \setminus \beta\, ,\\[7pt] &\pi(a {\textbf{u}})_j = 0\, , \quad j \in \gamma^c\, , \end{array} \right. \quad \iff \quad \left\{ \begin{array}{lll} &a u_j - \frac{|a{\textbf{u}}_\gamma| - 1}{|\gamma|}\geq x_j \, , \quad j \in \beta\, ,\\[7pt] &a u_j - \frac{|a{\textbf{u}}_\gamma| - 1}{|\gamma|} >0 \, , \quad j \in \gamma \setminus \beta\, ,\\[7pt] &a u_j \leq \frac{|a{\textbf{u}}_\gamma| - 1}{|\gamma|}\, , \quad j \in \gamma^c\, . \end{array} \right. \end{equation*}
   Then, if we write $a |{\textbf{u}}_\beta| = a - a |{\textbf{u}}_{\beta^c}|$ , the former conditions are equivalent to
   \begin{equation*} \left\{ \begin{array}{lll} &a u_j + \frac{a|{\textbf{u}}_{\gamma^c}|}{|\gamma|} \geq x_j + \frac{a-1}{|\gamma|} \, , \quad j \in \beta\, ,\\[7pt] &a u_j + \frac{a|{\textbf{u}}_{\gamma^c}|}{|\gamma|} > \frac{a-1}{|\gamma|} \, , \quad j \in \gamma \setminus \beta\, ,\\[7pt] &a u_j + \frac{a|{\textbf{u}}_{\gamma^c}|}{|\gamma|} \leq \frac{a-1}{|\gamma|}\, , \quad j \in \gamma^c\, . \end{array} \right. \end{equation*}
   We conclude the proof by writing ${\textbf{u}}_{\beta^c}$ = $|{\textbf{u}}_{\beta^c \setminus \{j\}}| + u_j$ for $j \in \gamma^c$ .

C.2. Proof of Proposition 1

This proof is a consequence of Lemma 1. Indeed for $r \geq 1$ , $t> 0$ , and $A \subset \mathbb{S}^{d-1}_+$ we have

\begin{align*}{\mathbb{P}} \big( \pi ({\textbf{X}} / t ) \in A, \, |{\textbf{X}}| / t > r &\mid |{\textbf{X}}| > t \big)\\[4pt] &= {\mathbb{P}} \big( \pi ({\textbf{X}} / t ) \in A \mid |{\textbf{X}}| / t > r \big) \frac{{\mathbb{P}}( |{\textbf{X}}| > tr)}{{\mathbb{P}}( |{\textbf{X}}| > t)}\\[4pt] &= {\mathbb{P}} \big( \pi (r {\textbf{X}} / (t r) ) \in A \mid |{\textbf{X}}| / t > r \big) {\mathbb{P}}( |{\textbf{X}}| > tr \mid |{\textbf{X}}| > t)\\[4pt] &= {\mathbb{P}} \big( r \pi_{1/r} ({\textbf{X}} / (t r) ) \in A \mid |{\textbf{X}}| > tr \big) {\mathbb{P}}( |{\textbf{X}}| > tr \mid |{\textbf{X}}| > t)\\[4pt] &= {\mathbb{P}} \big( r \pi_{1/r} ( \pi ({\textbf{X}}/ (t r) ) ) \in A \mid |{\textbf{X}}| > tr \big) {\mathbb{P}}( |{\textbf{X}}| > tr \mid |{\textbf{X}}| > t)\, ,\end{align*}

where the last equality results from Lemma 1. Then, when $t \to \infty$ , the continuity of $\pi_{1/r}$ and $\pi$ implies that

\begin{equation*}{\mathbb{P}}( {\textbf{Z}} \in A, \, Y > r) = {\mathbb{P}} ( r \pi_{1/r} \left({\textbf{Z}} \right) \in A ) {\mathbb{P}}(Y > r)\, ,\end{equation*}

and again applying Lemma 8.1 concludes the proof.

C.3. Proof of Corollary 1

Let ${\textbf{a}} \in \mathbb S^{d-1}_+$ be such that ${\mathbb{P}}( {\textbf{Z}} = {\textbf{a}} ) > 0$ , and define $\beta = \{j \,:\, a_j > 0\}$ . The goal is to prove that ${\textbf{a}} = {\textbf{e}}(\beta)/|\beta|$ .

Since ${\mathbb{P}}({\textbf{Z}} ={\textbf{a}} \mid Y > r) \to {\mathbb{P}}({\textbf{Z}} = {\textbf{a}})$ when $r \to 1$ , there exists $r_0 > 1$ such that ${\mathbb{P}}({\textbf{Z}} ={\textbf{a}} \mid Y > r) > 0$ for all $r \in (1, r_0)$ . Proposition 1 and Lemma 3 then imply that for all $r < r_0$ we have

\begin{equation*}0 < {\mathbb{P}}( \pi(r {\textbf{Z}}) = {\textbf{a}} )= {\mathbb{P}} \Big( Z_j - \frac{|{\textbf{Z}}_\beta|}{|\beta|} = \frac{a_j - 1/|\beta|}{r} \text{ for } j \in \beta, \, Z_j - \frac{|{\textbf{Z}}_\beta|}{|\beta|} \leq \frac{-1}{|\beta| r} \text{ for } j \in \beta^c \Big).\end{equation*}

If there exists j such that $a_j \neq 1/|\beta|$ , then the quantities

\begin{equation*}\frac{a_j - 1/|\beta|}{r}\end{equation*}

are all distinct when r varies in $(1, r_0)$ . This contradicts the fact that ${\textbf{Z}}$ has a discrete distribution. Hence we have $a_j = 1/|\beta|$ for all $j \in \beta$ .

C.4. Proof of Theorem 1

C.4.1. The distribution of ${\textbf{Z}}$ in terms of ${\boldsymbol{\Theta}}$

We consider $\beta \in {{\mathcal{P}}}_d^*$ and ${\textbf{x}} \in {{\mathcal{X}}}_\beta^0$ such that $x_j \neq 1/|\beta|$ for all $j \in \beta$ . We define the quantities $B_j = [(|\beta|\Theta_j - |{\boldsymbol{\Theta}}_\beta|) / (|\beta| x_j - 1)]_+$ for $j \in \beta$ . Then we obtain that

\begin{align*}G_\beta({\textbf{x}}) &= {\mathbb{P}}( \pi(Y {\boldsymbol{\Theta}})_\beta > {\textbf{x}}, \pi(Y {\boldsymbol{\Theta}})_{\beta^c}= {\textbf{0}}_{\beta^c})\\&= {\mathbb{P}}(Y \Theta_j - (Y|{\boldsymbol{\Theta}}_\beta|-1)/|\beta| > x_j \text{ for } j \in \beta, \,Y \Theta_j \leq (Y|{\boldsymbol{\Theta}}_\beta|-1) / |\beta| \text{ for } j \in \beta^c).\end{align*}

For $j \in \beta_+$ the condition $Y \Theta_j - (Y|{\boldsymbol{\Theta}}_\beta|-1)/|\beta| > x_j$ is equivalent to $B_j > 1/Y$ , whereas for $j \in \beta_-$ it is equivalent to $B_j < 1/Y$ . Moreover, for $j \in \beta^c$ the condition $Y \Theta_j \leq (Y|{\boldsymbol{\Theta}}_\beta|-1) / |\beta|$ is equivalent to $|{\boldsymbol{\Theta}}_\beta| - |\beta| \Theta_j \geq 1/Y$ . All in all we obtain that $G_\beta({\textbf{x}})$ is equal to

\begin{align*}{\mathbb{P}} \big(Y^{-\alpha} < B_j^\alpha \text{ for } j \in \beta_+,\,Y^{-\alpha} > B_j^\alpha \text{ for } j \in \beta_-,\,Y^{-\alpha} \leq (|{\boldsymbol{\Theta}}_\beta| - |\beta| \Theta_j)_+^\alpha\, \text{ for } j \in \beta^c\big)\, .\end{align*}

Since the random variable $Y^{-\alpha}$ follows a uniform distribution on (0,1) and is independent of ${\boldsymbol{\Theta}}$ , we obtain the desired result:

\begin{align*}G_\beta({\textbf{x}})&=\int_0^1{\mathbb{P}} \Big(u < \min _{j \in \beta_+} B_j^\alpha, \,u > \max_{j \in \beta_-} B_j^\alpha,\,u \leq \min_{j \in \beta^c}(|{\boldsymbol{\Theta}}_\beta| - |\beta| \Theta_j)_+^\alpha\Big)\, \mathrm d u\\&= \int_0^1{\mathbb{P}} \Big(\max_{j \in \beta_-} B_j^\alpha<u < \min _{j \in \beta_+} B_j^\alpha\wedge\min_{j \in \beta^c}(|{\boldsymbol{\Theta}}_\beta| - |\beta| \Theta_j)_+^\alpha\Big)\, \mathrm d u\\&= {\mathbb{E}} \Big[ \Big(1 \wedge\min _{j \in \beta_+} B_j^\alpha\wedge\min_{j \in \beta^c}(|{\boldsymbol{\Theta}}_\beta| - |\beta| \Theta_j)_+^\alpha- \max_{j \in \beta_-}B_j^\alpha\Big)_+ \Big]\, .\end{align*}

C.4.2. Equivalence of regular variation and sparse regular variation

The proof of this result is divided into two steps. The first one consists in characterizing regular variation via the convergence of ${\mathbb{P}}(|{\textbf{X}}|/t \leq 1+\epsilon, \, {\textbf{X}} / |{\textbf{X}}| \in A_{\textbf{x}} \mid |{\textbf{X}}| > t)$ when $t \to \infty$ and $\epsilon \to 0$ . This result is stated in the following lemma.

Lemma 4. Let ${\textbf{X}}$ be a random vector on $\mathbb{R}^d_+$ and $\alpha > 0$ . The following assumptions are equivalent:

1. (1) ${\textbf{X}}$ is regularly varying with tail index $\alpha$ .

2. 1. (2) a. $|{\textbf{X}}|$ is regularly varying with tail index $\alpha$ .

   2. b. For all $\beta \in {{\mathcal{P}}}_d^*$ and $\lambda_\beta$ -almost every ${\textbf{x}} \in {{\mathcal{X}}}_\beta$ the quantities

      (C.6) \begin{equation}(\alpha \epsilon)^{-1} \limsup_{t \to \infty} {\mathbb{P}}\big( |{\textbf{X}}| / t \leq 1+\epsilon, \, {\textbf{X}} / |{\textbf{X}}| \in A_{\textbf{x}} \mid |{\textbf{X}}| > t \big) \end{equation}
      and
      (C.7) \begin{equation}(\alpha \epsilon)^{-1} \liminf_{t \to \infty} {\mathbb{P}}\big( |{\textbf{X}}| / t \leq 1+\epsilon, \, {\textbf{X}} / |{\textbf{X}}| \in A_{\textbf{x}} \mid |{\textbf{X}}| > t \big) \end{equation}
      have a common limit $l(A_{\textbf{x}})$ when $\epsilon > 0$ converges to 0, and the function ${\textbf{x}} \mapsto l(A_{\textbf{x}})$ is continuous at ${\textbf{x}}$ .

In this case, l extends to a unique probability measure on ${{\mathcal{B}}}(\mathbb S^{d-1}_+)$ which coincides with the spectral measure of ${\textbf{X}}$ .

The second step then consists in proving that under Assumption (A) of Theorem 1 the second assumption of Lemma 4 holds.

Proof of Lemma 4. We first prove that (1) implies (2). If ${\textbf{X}} \in \mathrm{RV}(\alpha, {\boldsymbol{\Theta}})$ , then $|{\textbf{X}}|$ is regularly varying with index $\alpha$ , and the portmanteau theorem ensures that

(C.8) \begin{align} \lim_{t \to \infty} {\mathbb{P}}\big( |{\textbf{X}}| / t \leq 1+\epsilon, \, {\textbf{X}} / |{\textbf{X}}| \in A_{\textbf{x}} \mid |{\textbf{X}}| > t \big) &= {\mathbb{P}}( Y \leq 1 + \epsilon) {\mathbb{P}}({\boldsymbol{\Theta}} \in A_{\textbf{x}})\\ &= (1 - (1+\epsilon)^{-\alpha}) {\mathbb{P}}({\boldsymbol{\Theta}} \in A_{\textbf{x}}) \nonumber\, ,\end{align}

for $\epsilon > 0$ , $\beta \in {{\mathcal{P}}}_d^*$ , and ${\textbf{x}} \in {{\mathcal{X}}}_\beta$ such that ${\mathbb{P}}( {\boldsymbol{\Theta}} \in \partial A_{\textbf{x}}) = 0$ . Since ${\mathbb{P}}( {\boldsymbol{\Theta}} \in \partial A_{\textbf{x}}) \leq \sum_{i \in \beta} {\mathbb{P}}( \Theta_i=x_i )$ , the set of all ${\textbf{x}} \in {{\mathcal{X}}}_\beta$ for which the convergence (C.8) does not hold is at most countable and thus is $\lambda_\beta$ -negligible. After dividing both sides of (C.8) by $\alpha \epsilon$ and taking the limit when $\epsilon$ converges to 0 we obtain the convergence to ${\mathbb{P}}({\boldsymbol{\Theta}} \in A_{\textbf{x}})$ .

We now prove that (2) implies (1). We consider $\beta \in {{\mathcal{P}}}_d^*$ and denote by $\bar {{\mathcal{X}}}_\beta$ the set of all ${\textbf{x}} \in {{\mathcal{X}}}_\beta$ such that the common limit $l(A_{\textbf{x}})$ of (C.6) and (C.7) exists and is continuous. We also define $\bar {{\mathcal{X}}} = \{{\textbf{x}} \in B_+^d(0,1) \,:\, ({\textbf{x}}_\beta, {\textbf{0}}_{\beta^c}) \in {{\mathcal{X}}}_\beta \text{ for all } \beta \}$ .

For $\epsilon > 0$ and $u > 1$ we decompose the interval $(u,\infty)$ as follows:

\begin{equation*} (u,\infty) = \bigsqcup_{k=0}^\infty \big( u(1+\epsilon)^k, u(1+\epsilon)^{k+1} \big]\, , \end{equation*}

where $\bigsqcup$ denotes a disjoint union. Then we obtain, for ${\textbf{x}} \in \bar {{\mathcal{X}}}$ and $t > 0$ ,

(C.9) \begin{align} &{\mathbb{P}} \big(|{\textbf{X}}| / t > u, \, {\textbf{X}} / |{\textbf{X}}| \in A_{\textbf{x}} \mid |{\textbf{X}}| > t \big)\\ &= \sum_{k=0}^\infty {\mathbb{P}}\left( \frac{|{\textbf{X}}|}{tu(1+\epsilon)^k} \in (1,1+\epsilon],\, {\textbf{X}} / |{\textbf{X}}| \in A_{\textbf{x}} \; \middle| \; |{\textbf{X}}| > t \right) \nonumber\\ &= \sum_{k=0}^\infty {\mathbb{P}}\left( \frac{|{\textbf{X}}|}{tu(1+\epsilon)^k} \leq 1+\epsilon,\, {\textbf{X}} / |{\textbf{X}}| \in A_{\textbf{x}} \; \middle| \; |{\textbf{X}}| > t u(1+\epsilon)^k \right) \frac{{\mathbb{P}}( |{\textbf{X}}| > t u(1+\epsilon)^k )}{{\mathbb{P}}(|{\textbf{X}}| > t)}\, .\nonumber \end{align}

Fatou’s lemma and the fact that $|{\textbf{X}}|$ is regularly varying with tail index $\alpha$ together imply the two following inequalities:

\begin{align*} &\liminf_{t \to \infty} {\mathbb{P}} \big(|{\textbf{X}}| / t > u, \, {\textbf{X}} / |{\textbf{X}}| \in A_{\textbf{x}} \mid |{\textbf{X}}| > t \big)\\ &\geq \sum_{k=0}^\infty \big(u(1+\epsilon)^k\big)^{-\alpha} \liminf_{t \to \infty} {\mathbb{P}}\left( \frac{|{\textbf{X}}|}{tu(1+\epsilon)^k} \leq 1+\epsilon,\, {\textbf{X}} / |{\textbf{X}}| \in A_{\textbf{x}} \; \middle| \; \frac{|{\textbf{X}}|}{u(1+\epsilon)^k} > t \right)\\ &= \sum_{k=0}^\infty \big(u(1+\epsilon)^k\big)^{-\alpha} \liminf_{t \to \infty} {\mathbb{P}}\big( |{\textbf{X}}| / t \leq 1+ \epsilon, \,{\textbf{X}} / |{\textbf{X}}| \in A_{\textbf{x}} \mid |{\textbf{X}}| > t \big)\\ &= \frac{u^{-\alpha} }{1-(1+\epsilon)^{-\alpha}} \liminf_{t \to \infty} {\mathbb{P}}\big( |{\textbf{X}}| / t \leq 1+ \epsilon,\, {\textbf{X}} / |{\textbf{X}}| \in A_{\textbf{x}} \mid |{\textbf{{X}}}| > t \big)\, , \end{align*}

and

\begin{align*} &\limsup_{t \to \infty} {\mathbb{P}} \big(|{\textbf{X}}| / t > u, \, {\textbf{X}} / |{\textbf{X}}| \in A_{\textbf{x}} \mid |{\textbf{X}}| > t \big)\\ &\leq \sum_{k=0}^\infty \big(u(1+\epsilon)^k\big)^{-\alpha} \limsup_{t \to \infty} {\mathbb{P}}\left( \frac{|{\textbf{X}}|}{tu(1+\epsilon)^k} \leq 1+\epsilon,\, {\textbf{X}} / |{\textbf{X}}| \in A_{\textbf{x}} \; \middle| \; \frac{|{\textbf{X}}|}{u(1+\epsilon)^k} > t \right)\\ &= \sum_{k=0}^\infty \big(u(1+\epsilon)^k\big)^{-\alpha} \limsup_{t \to \infty} {\mathbb{P}}\big( |{\textbf{X}}| / t \leq 1+ \epsilon,\, {\textbf{X}} / |{\textbf{X}}| \in A_{\textbf{x}} \mid |{\textbf{X}}| > t \big)\\ &= \frac{u^{-\alpha} }{1-(1+\epsilon)^{-\alpha}} \limsup_{t \to \infty} {\mathbb{P}}\big( |{\textbf{X}}| / t \leq 1+ \epsilon, \,{\textbf{X}} / |{\textbf{X}}| \in A_{\textbf{x}} \mid |{\textbf{X}}| > t \big)\, . \end{align*}

Finally, we use Equations (C.6) and (C.7) and the relation $1-(1+\epsilon)^{-\alpha} \sim \alpha \epsilon$ , which imply that

\begin{align*} u^{-\alpha} l(A_{\textbf{x}}) &\leq \liminf_{t \to \infty} {\mathbb{P}} \big(|{\textbf{X}}| / t > u, \, {\textbf{X}} / |{\textbf{X}}| \in A_{\textbf{x}} \mid |{\textbf{X}}| > t \big)\\ &\leq \limsup_{t \to \infty} {\mathbb{P}} \big(|{\textbf{X}}| / t > u, \, {\textbf{X}} / |{\textbf{X}}| \in A_{\textbf{x}} \mid |{\textbf{X}}| > t \big) \leq u^{-\alpha} l(A_{\textbf{x}})\, . \end{align*}

This proves that

(C.10) \begin{equation}\lim_{t \to \infty} {\mathbb{P}} \big(|{\textbf{X}}| / t > u, \, {\textbf{X}} / |{\textbf{X}}| \in A \mid |{\textbf{X}}| > t \big) = u^{-\alpha} l( A )\, \end{equation}

for all $A = A_{\textbf{x}}$ such that ${\textbf{x}} \in \bar{{\mathcal{X}}}$ . The convergence also holds for $A = \mathbb S^{d-1}_+$ (resp. $A = \emptyset$ ) with $l(\mathbb S^{d-1}_+) = 1$ (resp. $l(\emptyset) = 0$ ). Then, by inclusion–exclusion we obtain the convergence of ${\mathbb{P}} \big({\textbf{X}} / |{\textbf{X}}| \leq {\textbf{x}} \mid |{\textbf{X}}| > t \big)$ for all ${\textbf{x}} \in \bar {{\mathcal{X}}}$ , and we denote by $F({\textbf{x}})$ this limit, which is continuous at any point ${\textbf{x}} \in \bar {{\mathcal{X}}}$ . In particular this implies the convergence ${\mathbb{P}}( X_j / |{\textbf{X}}| \leq x \mid |{\textbf{X}}| > t ) \to 1 - l(A_{x {\textbf{e}}_j}) \,=\!:\, L_j(x)$ for almost every $x \in (0, 1]$ . The functions $L_j$ are non-decreasing and continuous at almost every $x \in (0, 1]$ ; thus we extend it to a right-continuous function on [0,1]. Then F is continuous from above (see the definition on page 177 in [2]). For ${\textbf{a}} < {\textbf{b}} \in \bar {{\mathcal{X}}}$ we have the inequality ${\mathbb{P}} \big({\textbf{a}} < {\textbf{X}} / |{\textbf{X}}| \leq {\textbf{b}} \mid |{\textbf{X}}| > t \big) \geq 0$ , which implies that

(C.11) \begin{equation}\Delta_{({\textbf{a}}, {\textbf{b}}]} F \,:\!=\, \sum_{{\textbf{u}} \in {{\mathcal{V}}}({\textbf{a}}, {\textbf{b}})} \text{sign}({\textbf{u}}) F({\textbf{u}}) \geq 0\, , \end{equation}

where ${{\mathcal{V}}}({\textbf{a}}, {\textbf{b}})$ denotes all the vertices of $({\textbf{a}}, {\textbf{b}}]$ , $\text{sign}({\textbf{v}}) = 1$ if $u_k = a_k$ for an even number of values of k, and $\text{sign}({\textbf{v}}) = -1$ if $u_k = a_k$ for an odd number of values of k. Since F is continuous from above, the inequality (C.11) holds for all ${\textbf{a}} < {\textbf{b}} \in B_+^d(0,1)$ . Theorem 12.5 of [Reference Billingsley2] then ensures that there exists a measure S on $\mathbb{R}^d$ such that $S(({\textbf{a}}, {\textbf{b}}]) = \Delta_{({\textbf{a}}, {\textbf{b}}]} F$ . It is straightforward to see that S is a probability measure with support in $\mathbb S^{d-1}_+$ and that its restriction to $\mathbb S^{d-1}_+$ coincides with l. Thus Equation (C.10) can be rewritten as

\begin{equation*} {\mathbb{P}} \big(|{\textbf{X}}| / t > u, \, {\textbf{X}} / |{\textbf{X}}| \in \cdot \mid |{\textbf{X}}| > t \big) \stackrel{d}{\to} u^{-\alpha} l( \cdot )\, , \quad t \to \infty\, , \quad u > 1\, , \end{equation*}

and therefore proves that ${\textbf{X}}$ is regularly varying with tail index $\alpha$ and spectral measure $l(\cdot)$ .

Our aim is now to prove that the condition (2) holds for a sparsely regularly varying random vector which satisfies Assumption (A).

Proof of Theorem 1.

We consider a random vector ${\textbf{X}} \in \mathrm{SRV}(\alpha, {\textbf{Z}})$ which satisfies Assumption (A). For $\beta \in {{\mathcal{P}}}_d^*$ and ${\textbf{x}} \in {{\mathcal{X}}}_\beta$ we decompose the probability ${\mathbb{P}}\big( |{\textbf{X}}| / t \leq 1+\epsilon, \, {\textbf{X}} / |{\textbf{X}}| \in A_{\textbf{x}} \mid |{\textbf{X}}| > t \big)$ as follows:

\begin{equation*}\sum_{\gamma \supset \beta}{\mathbb{P}}\big( |{\textbf{X}}| / t \leq 1+\epsilon, \, {\textbf{X}} / |{\textbf{X}}| \in A_{\textbf{x}},\, \pi({\textbf{X}}/t) \in C_\gamma \mid |{\textbf{X}}| > t \big)\, ,\end{equation*}

where the restriction of the sum to the directions $\gamma \supset \beta$ is a consequence of the first part of Lemma 2. Moreover, for $\gamma \supset \beta$ , Lemma 8.1 then ensures that the probability ${\mathbb{P}}\big( |{\textbf{X}}| / t \leq 1+\epsilon, \, {\textbf{X}} / |{\textbf{X}}| \in A_{\textbf{x}},\, \pi({\textbf{X}}/t) \in C_\gamma \mid |{\textbf{X}}| > t \big)$ is bounded above by

(C.12) \begin{equation}{\mathbb{P}}\big( |{\textbf{X}}| / t \leq 1+\epsilon, \, \pi({\textbf{X}}/t) \in A_{{\textbf{x}} - \epsilon / |\gamma|} \cap C_\gamma \mid |{\textbf{X}}| > t \big)\, ,\end{equation}

assuming that $\epsilon > 0$ is small enough so that the vector ${\textbf{x}} - \epsilon / |\gamma|$ has positive coordinates, and bounded below by

(C.13) \begin{equation}{\mathbb{P}}\big( |{\textbf{X}}| / t \leq 1+\epsilon, \, \pi({\textbf{X}}/t) \in A_{{\textbf{x}} (1+\epsilon)} \cap C_\gamma \mid |{\textbf{X}}| > t \big)\, .\end{equation}

In order to deal simultaneously with both probabilities in (C.12) and (C.13), we write

(C.14) \begin{equation}{\mathbb{P}}\big( |{\textbf{X}}| / t \leq 1+\epsilon, \, \pi({\textbf{X}}/t) \in A_{\psi_\epsilon({\textbf{x}})} \cap C_\gamma \mid |{\textbf{X}}| > t \big)\, ,\end{equation}

where $\psi_\epsilon({\textbf{x}})$ is equal either to ${\textbf{x}} - \epsilon / |\gamma|$ or to ${\textbf{x}} (1+\epsilon)$ .

The proof is then divided into two steps. The first step consists in proving that these two probabilities (C.12) and (C.13) converge when $t\to \infty$ for $\lambda_\beta$ -almost every ${\textbf{x}} \in {{\mathcal{X}}}_\beta$ . Then the goal is to prove that after division by $\epsilon$ these two limits converge to the same quantity when $\epsilon \to 0$ . We extract the first step as a separate lemma.

Lemma 5. For all $\beta \subset \gamma \in {{\mathcal{P}}}_d^*$ , for $\lambda_\beta$ -almost every ${\textbf{x}} \in {{\mathcal{X}}}_\beta$ , for almost every $\epsilon > 0$ we have the following convergence:

(C.15) \begin{equation}\lim \limits_{t \to \infty} {\mathbb{P}}\big( \pi({\textbf{X}}/t) \in A_{\psi_\epsilon({\textbf{x}})} \cap C_\gamma, \, |{\textbf{X}}| / t > 1+\epsilon \mid |{\textbf{X}}| > t \big) = {\mathbb{P}}( {\textbf{Z}} \in A_{\psi_\epsilon({\textbf{x}})} \cap C_\gamma, \, Y > 1+\epsilon) \, . \end{equation}

Proof. We use Proposition 1, which implies that the right-hand side in (C.15) is equal to

\begin{align*} {\mathbb{P}}( {\textbf{Z}} \in A_{\psi_\epsilon({\textbf{x}})} \cap C_\gamma, \, Y > 1+\epsilon) &= {\mathbb{P}}( {\textbf{Z}} \in A_{\psi_\epsilon({\textbf{x}})} \cap C_\gamma \mid Y > 1+\epsilon) {\mathbb{P}}(Y > 1+\epsilon)\\ &= {\mathbb{P}}( \pi((1+\epsilon) {\textbf{Z}}) \in A_{\psi_\epsilon({\textbf{x}})} \cap C_\gamma) (1+\epsilon)^{-\alpha}\, . \end{align*}

The left-hand side in (C.15) can be rewritten as follows:

\begin{align*} {\mathbb{P}}\big( \pi({\textbf{X}}/t) \in &A_{\psi_\epsilon({\textbf{x}})} \cap C_\gamma, \, |{\textbf{X}}| / t > 1+\epsilon \mid |{\textbf{X}}| > t \big)\\ &= {\mathbb{P}}\big( \pi({\textbf{X}}/t) \in A_{\psi_\epsilon({\textbf{x}})} \cap C_\gamma \mid |{\textbf{X}}| > t (1+\epsilon) \big) \frac{{\mathbb{P}}(|{\textbf{X}}| > t (1+\epsilon) )}{{\mathbb{P}}(|{\textbf{X}}| > t)}\, . \end{align*}

The ratio ${\mathbb{P}}(|{\textbf{X}}| > t (1+\epsilon)) / {\mathbb{P}}(|{\textbf{X}}| > t)$ converges to $ (1+\epsilon)^{-\alpha}$ when $t \to \infty$ since $|{\textbf{X}}|$ is regularly varying with tail index $\alpha$ . Furthermore, the probability ${\mathbb{P}}\big( \pi({\textbf{X}}/t) \in A_{\psi_\epsilon({\textbf{x}})} \cap C_\gamma \mid |{\textbf{X}}| > t (1+\epsilon) \big)$ converges when $t \to \infty$ if and only if ${\mathbb{P}}\big( \pi( (1+\epsilon) {\textbf{X}}/t) \in A_{\psi_\epsilon({\textbf{x}})} \cap C_\gamma \mid |{\textbf{X}}| > t \big)$ converges when $t \to \infty$ , and then both probabilities have the same limit. We use Lemma 8.1 and the relation $\pi_z({\textbf{v}}) = z \pi({\textbf{v}}/z)$ for ${\textbf{v}} \in \mathbb{R}^d_+$ and $z > 0$ , which imply that

\begin{equation*} \pi( (1+\epsilon) {\textbf{X}}/t) = (1+\epsilon) \pi_{1/ (1+\epsilon)} ({\textbf{X}}/t) = (1+\epsilon) \pi_{1/ (1+\epsilon)} ( \pi({\textbf{X}}/t) ) = \pi ( (1+\epsilon)\pi({\textbf{X}}/t) )\, . \end{equation*}

Hence the convergence in Equation (C.15) holds if and only if

(C.16) \begin{equation}{\mathbb{P}}\big( \pi ( (1+\epsilon) \pi({\textbf{X}}/t) ) \in A_{\psi_\epsilon({\textbf{x}})} \cap C_\gamma \mid |{\textbf{X}}| > t \big) \to {\mathbb{P}}( \pi( (1+\epsilon){\textbf{Z}}) \in A_{\psi_\epsilon({\textbf{x}})} \cap C_\gamma)\, , \quad t \to \infty \, . \end{equation}

The equivalence (C.5) then implies that the former convergence holds if and only if the probability

\begin{align*} {\mathbb{P}}\bigg( \phi_\gamma(\pi({\textbf{X}}/t))_\beta \geq \frac{\psi_\epsilon({\textbf{x}})}{1+\epsilon} + \bigg.&\frac{\epsilon}{(1+\epsilon) |\gamma|}, \, \min_{i \in \gamma \setminus \beta} \phi_\gamma(\pi({\textbf{X}}/t))_i > \frac{\epsilon}{(1+\epsilon) |\gamma|}, \, \\ &\left.\max_{j \in \gamma^c} \phi_\gamma(\pi({\textbf{X}}/t))_j \leq \frac{\epsilon}{(1+\epsilon) (|\gamma|+1)} \; \middle| \; |{\textbf{X}}| > t \right) \end{align*}

converges to

\begin{align*} {\mathbb{P}}\bigg( \phi_\gamma({\textbf{Z}})_\beta \geq \frac{\psi_\epsilon({\textbf{x}})}{1+\epsilon} + \frac{\epsilon}{(1+\epsilon)|\gamma|}, \, \bigg.&\min_{i \in \gamma \setminus \beta} \phi_\gamma({\textbf{Z}})_i > \frac{\epsilon}{(1+\epsilon) |\gamma|}, \,\\ &\bigg. \max_{j \in \gamma^c} \phi_\gamma({\textbf{Z}})_j \leq \frac{\epsilon}{(1+\epsilon) (|\gamma|+1)} \bigg)\, \end{align*}

when $t \to \infty$ .

For $\gamma \supset \beta$ , the set of all $\epsilon > 0$ for which

\begin{equation*}{\mathbb{P}}\Big( \min_{i \in \gamma \setminus \beta} \phi_\gamma({\textbf{Z}})_i = \frac{\epsilon}{(1+\epsilon)|\gamma|}\Big) > 0\end{equation*}

or for which

\begin{equation*}{\mathbb{P}}\Big( \max_{j \in \gamma^c} \phi_\gamma({\textbf{Z}})_j = \frac{\epsilon}{(1+\epsilon)(|\gamma|+1)}\Big) > 0\end{equation*}

is at most countable, so that for almost every $\epsilon > 0$ we have

\begin{equation*} {\mathbb{P}}\Big( \min_{i \in \gamma \setminus \beta} \phi_\gamma({\textbf{Z}})_i = \frac{\epsilon}{(1+\epsilon) |\gamma|} \Big) = {\mathbb{P}}\Big( \max_{j \in \gamma^c} \phi_\gamma({\textbf{Z}})_j = \frac{\epsilon}{(1+\epsilon) (|\gamma|+1)} \Big) = 0\, \end{equation*}

for all $\gamma \supset \beta$ . Let us denote by ${{\mathcal{E}}}_\beta$ the set of all these $\epsilon >0$ . The same argument implies that for $\lambda_\beta$ -almost every ${\textbf{x}} \in {{\mathcal{X}}}_\beta$ we have

\begin{equation*} {\mathbb{P}}\Big( \phi_\gamma({\textbf{Z}})_j = \frac{\psi_\epsilon({\textbf{x}})_j}{1+\epsilon} + \frac{\epsilon}{(1+\epsilon)|\gamma|} \Big) = 0 \, \quad \text{for all } j \in \beta\, , \quad \gamma \supset \beta, \, \text{ and } \epsilon \in {{\mathcal{E}}}_\beta\, . \end{equation*}

Let us denote by $\bar{{\mathcal{X}}}_\beta$ the set of all these ${\textbf{x}}$ . We have proved that for all $\epsilon \in {{\mathcal{E}}}_\beta$ and ${\textbf{x}} \in \bar{{\mathcal{X}}}_\beta$ we have the convergence

\begin{equation*} \lim \limits_{t \to \infty} {\mathbb{P}}\big( \pi({\textbf{X}}/t) \in A_{\psi_\epsilon({\textbf{x}})} \cap C_\gamma, \, |{\textbf{X}}| / t > 1+\epsilon \mid |{\textbf{X}}| > t \big) = {\mathbb{P}}( {\textbf{Z}} \in A_{\psi_\epsilon({\textbf{x}})} \cap C_\gamma, \, Y > 1+\epsilon) \, . \end{equation*}

This concludes the proof of the Lemma 5.

End of the proof of Theorem 1.

We use the same notation $\bar {{\mathcal{X}}}_\beta$ and ${{\mathcal{E}}}_\beta$ as in the proof of Lemma. We start with the following decomposition:

\begin{align*}{\mathbb{P}}\big( \pi ( {\textbf{X}}/t ) &\in A_{\psi_\epsilon({\textbf{x}})} \cap C_\gamma, \, |{\textbf{X}}|/t \leq 1+\epsilon \mid |{\textbf{X}}| > t \big)\\&={\mathbb{P}}\big( \pi ( {\textbf{X}}/t ) \in A_{\psi_\epsilon({\textbf{x}})} \cap C_\gamma, \, |{\textbf{X}}|/t \leq 1+\epsilon^2 \mid |{\textbf{X}}| > t \big)\\&+{\mathbb{P}}\big( \pi ( {\textbf{X}}/t ) \in A_{\psi_\epsilon({\textbf{x}})} \cap C_\gamma, \, |{\textbf{X}}|/t \in (1+\epsilon^2, 1+\epsilon] \mid |{\textbf{X}}| > t \big)\, .\end{align*}

The introduction of the term with $\epsilon^2$ on the right-hand side follows from the fact that the probability ${\mathbb{P}}\big( \pi ( {\textbf{X}}/t ) \in A_{\psi_\epsilon({\textbf{x}})} \cap C_\gamma \mid |{\textbf{X}}| > t \big)$ may not converge. The latter probability is bounded by

\begin{equation*}{\mathbb{P}}\big( |{\textbf{X}}|/t \leq 1+\epsilon^2 \mid |{\textbf{X}}| > t \big) \to 1 - (1+\epsilon^2)^{-\alpha} = o(\epsilon).\end{equation*}

Moreover, Lemma 5 ensures that

\begin{align*}{\mathbb{P}}\big( \pi ( {\textbf{X}}/t ) \in A_{\psi_\epsilon({\textbf{x}})} \cap C_\gamma, \, |{\textbf{X}}|/t &\in (1+\epsilon^2, 1+\epsilon] \mid |{\textbf{X}}| > t \big)\\&\to{\mathbb{P}}\big( {\textbf{Z}} \in A_{\psi_\epsilon({\textbf{x}})} \cap C_\gamma, \, Y \in (1+\epsilon^2, 1+\epsilon] \big)\, ,\end{align*}

when $t \to \infty$ for $\epsilon, \epsilon^2 \in {{\mathcal{E}}}_\beta$ and ${\textbf{x}} \in \bar{{\mathcal{X}}}_\beta$ . Hence we obtain

\begin{align*}\lim_{t \to \infty} &{\mathbb{P}}\big( \pi ( {\textbf{X}}/t ) \in A_{\psi_\epsilon({\textbf{x}})} \cap C_\gamma, \, |{\textbf{X}}|/t \leq 1+\epsilon \mid |{\textbf{X}}| > t \big)\\&=\lim_{t \to \infty}{\mathbb{P}}\big( \pi ( {\textbf{X}}/t ) \in A_{\psi_\epsilon({\textbf{x}})} \cap C_\gamma, \, |{\textbf{X}}|/t \in (1+\epsilon^2, 1+\epsilon] \mid |{\textbf{X}}| > t \big) + o(\epsilon)\\&={\mathbb{P}}\big( {\textbf{Z}} \in A_{\psi_\epsilon({\textbf{x}})} \cap C_\gamma, \, Y \in (1+\epsilon^2, 1+\epsilon] \big) + o(\epsilon)\\&={\mathbb{P}}\big( {\textbf{Z}} \in A_{\psi_\epsilon({\textbf{x}})} \cap C_\gamma, \, Y \leq 1+\epsilon \big) + o(\epsilon)\, .\end{align*}

Then Proposition 1 implies that

(C.17) \begin{align}{\mathbb{P}}\big( {\textbf{Z}}_\beta &\in A_{\psi_\epsilon({\textbf{x}})} \cap C_\gamma, \, Y \leq 1+\epsilon \big) \nonumber\\[2pt] &= {\mathbb{P}}\big( {\textbf{Z}} \in A_{\psi_\epsilon({\textbf{x}})} \cap C_\gamma\big)- {\mathbb{P}}\big( {\textbf{Z}}\in A_{\psi_\epsilon({\textbf{x}})} \cap C_\gamma \mid Y > 1+\epsilon \big) (1+\epsilon)^{-\alpha} \nonumber\\[2pt] &= {\mathbb{P}}\big( {\textbf{Z}} \in A_{\psi_\epsilon({\textbf{x}})} \cap C_\gamma \big)- {\mathbb{P}}\big( \pi((1+\epsilon){\textbf{Z}}) \in A_{\psi_\epsilon({\textbf{x}})} \cap C_\gamma \big) (1+\epsilon)^{-\alpha} \nonumber\\[2pt] &= \big[1 - (1+\epsilon)^{-\alpha}\big]{\mathbb{P}}\big( {\textbf{Z}} \in A_{\psi_\epsilon({\textbf{x}})} \cap C_\gamma \big) \end{align}

(C.18) \begin{align}& + (1+\epsilon)^{-\alpha} \big[{\mathbb{P}}\big( {\textbf{Z}} \in A_{\psi_\epsilon({\textbf{x}})} \cap C_\gamma \big)-{\mathbb{P}}\big( \pi((1+\epsilon){\textbf{Z}})\in A_{\psi_\epsilon({\textbf{x}})} \cap C_\gamma \big) \big]\, .\end{align}

The sets $A_{\psi_\epsilon({\textbf{x}})}$ decrease when $\epsilon \to 0$ and satisfy $\cap_{\epsilon >0} A_{\psi_\epsilon({\textbf{x}})} = A_{\textbf{x}}$ . Hence the term in (C.17) divided by $\epsilon$ converges to $\alpha {\mathbb{P}}( {\textbf{Z}} \in A_{\textbf{x}} \cap C_\gamma)$ . For the term in (C.18) we use (C.5), which states that the event $\{ \pi((1+\epsilon){\textbf{Z}}) \in A_{\psi_\epsilon({\textbf{x}})} \cap C_\gamma \}$ corresponds to

\begin{equation*}\left\{\begin{array}{lll}&\phi_\gamma({\textbf{Z}})_j \geq \frac{\psi_\epsilon({\textbf{x}})_j}{(1+\epsilon)} + \frac{\epsilon}{|\gamma| (1+\epsilon)}\, , \quad j \in \beta \, ,\\[8pt] &\min_{j \in \gamma \setminus \beta} \phi_\gamma({\textbf{Z}})_j > \frac{\epsilon}{|\gamma|(1+\epsilon)}\, ,\\[8pt] &\max_{j \in \gamma^c} \phi_\gamma({\textbf{Z}})_j \leq \frac{\epsilon}{(|\gamma|+1)(1+\epsilon)}\, .\end{array}\right.\end{equation*}

Hence the difference of probabilities in (C.18) corresponds to the difference

\begin{equation*}H_{\beta, \gamma} \big( \psi_\epsilon({\textbf{x}}), 0, 0 \big)-H_{\beta, \gamma} \Big(\frac{\psi_\epsilon({\textbf{x}})}{(1+\epsilon)} + \frac{\epsilon}{|\gamma| (1+\epsilon)},\frac{\epsilon}{|\gamma|(1+\epsilon)},\frac{\epsilon}{(|\gamma|+1)(1+\epsilon)}\Big)\, .\end{equation*}

After a division by $\epsilon$ this difference converges to

\begin{equation*}\mathrm d H_{\beta, \gamma} ({\textbf{x}}_\beta, 0, 0)\cdot({\textbf{x}}_\beta -1/|\gamma|, -1/|\gamma| , -1/(|\gamma|+1))\, ,\quad \epsilon \to 0\, ,\end{equation*}

for $\psi_\epsilon({\textbf{x}}) = {\textbf{x}}/(1+\epsilon)$ and $\psi_\epsilon({\textbf{x}}) = {\textbf{x}} - \epsilon/|\gamma|$ .

All in all, we have proved that for $\lambda_\beta$ -almost every ${\textbf{x}} \in {{\mathcal{X}}}_\beta$ , both limits

\begin{equation*}\lim_{\epsilon \to 0}\liminf_{t \to \infty}\epsilon^{-1} {\mathbb{P}}\big( |{\textbf{X}}| / t \leq 1+\epsilon, \, {\textbf{X}} / |{\textbf{X}}| \in A_{\textbf{x}} \mid |{\textbf{X}}| > t \big)\end{equation*}

and

\begin{equation*}\lim_{\epsilon \to 0}\limsup_{t \to \infty}\epsilon^{-1} {\mathbb{P}}\big( |{\textbf{X}}| / t \leq 1+\epsilon, \, {\textbf{X}} / |{\textbf{X}}| \in A_{\textbf{x}} \mid |{\textbf{X}}| > t \big)\end{equation*}

exist and are equal to

\begin{align*}l(A_{\textbf{x}})&=\sum_{\gamma \supset \beta}\alpha {\mathbb{P}}({\textbf{Z}} \in A_{\textbf{x}} \cap C_\gamma)+ \sum_{\gamma \supset \beta} \mathrm d H_{\beta, \gamma} ({\textbf{x}}_\beta, 0, 0) \cdot \Big({\textbf{x}}_\beta - \frac{1}{|\gamma|}, -\frac{1}{|\gamma|}, -\frac{1}{|\gamma|+1} \Big)\\&= \alpha {\mathbb{P}}({\textbf{Z}} \in A_{\textbf{x}})+ \sum_{\gamma \supset \beta} \mathrm d H_{\beta, \gamma} ({\textbf{x}}_\beta, 0, 0) \cdot \Big({\textbf{x}}_\beta - \frac{1}{|\gamma|}, -\frac{1}{|\gamma|}, -\frac{1}{|\gamma|+1} \Big)\, .\end{align*}

Then Assumption (A) ensures that for $\lambda_\beta$ -almost every ${\textbf{x}} \in {{\mathcal{X}}}_\beta$ , the function ${\textbf{x}} \mapsto l(A_{\textbf{x}})$ is continuous at ${\textbf{x}}$ , and Lemma 4 allows us to conclude.

C.5. Proof of Proposition 2

Recall that we have defined the quantities $\theta_{\beta, \, i} = \sum_{j \in \beta} (\theta_j - \theta_i)$ and $\theta_{\beta, \, i, \, +} = \sum_{j \in \beta} (\theta_j - \theta_i)_+$ for ${\boldsymbol \theta} \in \mathbb S^{d-1}_+$ , $\beta \in {{\mathcal{P}}}_d^*$ , and $1 \leq i \leq d$ .

1. We prove only the convergence (3.2) (the proof of (3.3) is similar). For $\beta \in {{\mathcal{P}}}_d^*$ , Lemma 3 ensures that $\pi(Y {\boldsymbol{\Theta}})\in C_\beta$ if and only if $\max_{i \in \beta} \Theta_{\beta, \, i} < 1/Y$ and $\min_{i \in \beta^c} \Theta_{\beta, \, i} \geq 1/Y$ . Hence (3.2) is equivalent to

(C.19) \begin{equation} {\mathbb{P}}\big( (|{\textbf{X}}|/t, {\textbf{X}}/|{\textbf{X}}|) \in D_\beta \mid |{\textbf{X}}|>t \big) \to {\mathbb{P}}((Y,{\boldsymbol{\Theta}}) \in D_\beta)\, ,\end{equation}

with

\begin{equation*}D_\beta = \big\{ (r, \theta) \in (1,\infty) \times \mathbb S^{d-1}_+ \,:\, \theta_{\beta, \, i} < 1/r \text{ for } i \in \beta, \text{ and } \theta_{\beta, \, i} \geq 1/r \text{ for } i \in \beta^c \big\}\, ,\end{equation*}

and this convergence holds if ${\mathbb{P}}((Y,{\boldsymbol{\Theta}}) \in \partial D_\beta) = 0$ . The boundary $\partial D_\beta$ satisfies the inequality

\begin{align*}{\mathbb{P}}((Y,{\boldsymbol{\Theta}}) \in \partial D_\beta)\leq \sum_{i = 1}^d {\mathbb{P}} ( \Theta_{\beta, \, i} = Y^{-1} )\, ,\end{align*}

and all the terms of the sum are null since Y is a continuous random variable independent of ${\boldsymbol{\Theta}}$ . Thus ${\mathbb{P}}((Y,{\boldsymbol{\Theta}}) \in \partial D_\beta) = 0$ , which implies that the convergence (C.19) holds and then that the convergence (3.2) holds as well.

2. Following Lemma 3, the probability that ${\textbf{Z}}$ belongs to $C_\beta$ is equal to

\begin{align*}{\mathbb{P}}({\textbf{Z}} \in C_\beta)&= {\mathbb{P}} \Big( \max_{j \in \beta} \sum_{k \in \beta} (Y \Theta_k - Y \Theta_j) < 1, \, \min_{j \in \beta^c} \sum_{k\in \beta} (Y \Theta_k - Y \Theta_j) \geq 1 \Big)\\&= {\mathbb{P}} \Big( \max_{j \in \beta} \, \Theta_{\beta, \, j, \, +}^\alpha < Y^{-\alpha}, \, \min_{j \in \beta^c} \, \Theta_{\beta, \, j, \, +}^\alpha \geq Y^{-\alpha} \Big)\\&= \int_0^1 {\mathbb{P}} \Big( \max_{j \in \beta} \Theta_{\beta, \, j, \, +}^\alpha < u \leq \min_{j \in \beta^c} \Theta_{\beta, \, j, \, +}^\alpha \Big) \; \mathrm d u\\&= {\mathbb{E}} \Big[ \Big( \min_{j \in \beta^c} \Theta_{\beta, \, j, \, +}^\alpha - \max_{j \in \beta} \Theta_{\beta, \, j, \, +}^\alpha \Big)_+ \Big]\, .\end{align*}

Similarly, Equation (3.5) is also a consequence of Lemma 3 since we have the relations

(C.20) \begin{align}{\mathbb{P}}({\textbf{Z}}_{\beta^c} = 0)&= {\mathbb{P}}\big( 1 \leq \min_{j \in \beta^c} \sum_{k=1}^d (Y\Theta_k - Y\Theta_j)_+\big)= {\mathbb{P}}\big(Y^{-\alpha} \leq \min_{j \in \beta^c} \Theta_{\beta, \, j, \, +}^\alpha \big)\\[4pt] &= \int_0^1 {\mathbb{P}}\big(u \leq \min_{j \in \beta^c} \Theta_{\beta, \, j, \, +}^\alpha \big) \, \mathrm d u= {\mathbb{E}} \big[ \min_{j \in \beta^c} \Theta_{\beta, \, j, \, +}^\alpha \big] \, . \nonumber\end{align}

This proves (3.5) and concludes the proof of the proposition.

C.6. Proof of Theorem 2

We first state an inequality which will be used to prove both results of Theorem 2.

Lemma 6. For $\beta \in {{\mathcal{P}}}_d^*$ we have the inequality

(C.21) \begin{equation} {\mathbb{P}}({\boldsymbol{\Theta}} \in C_\beta) \leq {\mathbb{P}} \big( \max_{j \in \beta} \Theta_{\beta, \, j, \, +} < 1 \big) \, . \end{equation}

Proof of Lemma 6. The relation in (C.21) is equivalent to

\begin{equation*}{\mathbb{P}} \big( \max_{j \in \beta} \Theta_{\beta, \, j, \, +} = 1 \big) \leq {\mathbb{P}}({\boldsymbol{\Theta}} \notin C_\beta)\, .\end{equation*}

The probability on the left-hand side can be rewritten as

\begin{equation*}{\mathbb{P}} \big( \max_{j \in \beta} \Theta_{\beta, \, j, \, +} = 1 \big) = {\mathbb{P}} \Big( \sum_{k \in \beta} (\Theta_k - \min_{j \in \beta} \Theta_j) = 1 \Big) = {\mathbb{P}} \Big( \sum_{k \in \beta} \Theta_k = 1 + |\beta| \min_{j \in \beta} \Theta_j \Big)\, .\end{equation*}

Since ${\boldsymbol{\Theta}} \in \mathbb S^{d-1}_+$ , the equality $\sum_{k \in \beta} \Theta_k = 1 + |\beta| \min_{j \in \beta} \Theta_j$ holds only if there exists $k \in \beta$ such that $\Theta_k = 0$ . Thus we obtain the inequality

\begin{equation*} {\mathbb{P}} \Big( \max_{j \in \beta} \Theta_{\beta, \, j, \, +} = 1 \Big) \leq {\mathbb{P}}( \Theta_k = 0 \text{ for some } k \in \beta) \leq {\mathbb{P}}( {\boldsymbol{\Theta}} \notin C_\beta )\, ,\end{equation*}

which concludes the proof.

We now move on to the proof of Theorem 2. The proof of these results relies on Equation (3.4) and Lemma 6 above.

1. If $\beta \in {{\mathcal{P}}}_d^*$ is such that ${\mathbb{P}}({\boldsymbol{\Theta}} \in C_\beta) > 0$ , then Equation (3.4) implies that

(C.22) \begin{align}{\mathbb{P}}( {\textbf{Z}} \in C_\beta )& \geq {\mathbb{E}} \Big[ \Big(\min_{j \in \beta^c} \Theta_{\beta, \, j, \, +}^\alpha - \max_{j \in \beta} \Theta_{\beta, \, j, \, +}^\alpha\Big)_+ {1}_{ {\boldsymbol{\Theta}} \in C_\beta }\Big] \nonumber\\[4pt] & = {\mathbb{E}} \Big[ \Big(1 - \max_{j \in \beta} \Theta_{\beta, \, j, \, +}^\alpha\Big) {1}_{ {\boldsymbol{\Theta}} \in C_\beta } \Big] \nonumber\\[4pt] & = {\mathbb{E}} \Big[ \Big(1 - \max_{j \in \beta} \Theta_{\beta, \, j, \, +}^\alpha\Big) \mid {\boldsymbol{\Theta}} \in C_\beta\Big] {\mathbb{P}}({\boldsymbol{\Theta}} \in C_\beta)\, . \end{align}

The expectation is positive by Lemma 6, and the probability ${\mathbb{P}}({\boldsymbol{\Theta}} \in C_\beta)$ is positive by assumption. This shows that ${\mathbb{P}}({\textbf{Z}} \in C_\beta) > 0$ .

2. We prove the two implications separately.

We first consider a maximal direction $\beta$ of ${\boldsymbol{\Theta}}$ . The first part of the theorem ensures that ${\mathbb{P}}({\textbf{Z}} \in C_\beta) > 0$ . In addition, if $\beta' \supsetneq \beta$ , then Equation (3.8) gives

\begin{equation*}{\mathbb{P}}({\textbf{Z}} \in C_{\beta'}) \leq {\mathbb{P}}({\textbf{Z}}_{\beta'} > 0) \leq {\mathbb{P}}({\boldsymbol{\Theta}}_{\beta'} > 0)\, ,\end{equation*}

and this last probability equals zero since $\beta$ is a maximal direction for ${\boldsymbol{\Theta}}$ . This proves that $\beta$ is a maximal direction for ${\textbf{Z}}$ .

We now consider a maximal direction $\beta$ of ${\textbf{Z}}$ . We claim that for $\beta' \supsetneq \beta$ , ${\mathbb{P}}({\boldsymbol{\Theta}} \in C_{\beta'}) = 0$ . If not, the first part of the theorem implies that ${\mathbb{P}}({\textbf{Z}} \in C_{\beta'}) > 0$ , which contradicts the maximality of $\beta$ for ${\textbf{Z}}$ .

Secondly, Equation (3.4) implies that

(C.23) \begin{align} {\mathbb{P}}({\textbf{Z}} \in C_\beta)&= {\mathbb{E}} \Big[ \Big(\min_{j \in \beta^c} \Theta_{\beta, \, j, \, +}^\alpha- \max_{j \in \beta} \Theta_{\beta, \, j, \, +}^\alpha\Big)_+ \Big] \nonumber\\&= {\mathbb{E}} \Big[ \Big(\min_{j \in \beta^c} \Theta_{\beta, \, j, \, +}^\alpha- \max_{j \in \beta} \Theta_{\beta, \, j, \, +}^\alpha\Big)_+ {1}_{{\boldsymbol{\Theta}} \in C_\beta} \Big]\end{align}

(C.24) \begin{align}&+ {\mathbb{E}} \Big[ \Big(\min_{j \in \beta^c} \Theta_{\beta, \, j, \, +}^\alpha- \max_{j \in \beta} \Theta_{\beta, \, j, \, +}^\alpha\big)_+ {1}_{{\boldsymbol{\Theta}} \notin C_\beta} \Big] \nonumber\\&= E_1 + E_2\, .\end{align}

The first term $E_1$ has already been calculated in (C.22). It is equal to

\begin{equation*}E_1 = {\mathbb{E}} \Big[1 - \max_{j \in \beta} \Theta_{\beta, \, j, \, +}^\alpha \mid {\boldsymbol{\Theta}} \in C_\beta\Big] {\mathbb{P}}( {\boldsymbol{\Theta}} \in C_\beta)\, .\end{equation*}

For the second term $E_2$ , the assumption ${\boldsymbol{\Theta}} \notin C_\beta$ implies that there exists $l \in \beta$ such that $\Theta_l = 0$ , or that there exists $r \in \beta^c$ such that $\Theta_r >0$ . We then decompose $E_2$ into two terms:

\begin{align*}&{\mathbb{E}} \Big[ \Big(\min_{j \in \beta^c} \Theta_{\beta, \, j, \, +}^\alpha - \max_{j \in \beta} \Theta_{\beta, \, j, \, +}^\alpha\Big)_+ {1}_{{\boldsymbol{\Theta}} \notin C_\beta} \Big] \\[5pt] &\leq {\mathbb{E}} \Big[ \Big(\min_{j \in \beta^c} \Theta_{\beta, \, j, \, +}^\alpha - \max_{j \in \beta} \Theta_{\beta, \, j, \, +}^\alpha\Big)_+ {1}_{\exists l \in \beta, \, \Theta_l =0} \Big] \\[5pt] &+ {\mathbb{E}}\Big[ \Big(\min_{j \in \beta^c} \Theta_{\beta, \, j, \, +}^\alpha - \max_{j \in \beta} \Theta_{\beta, \, j, \, +}^\alpha\Big)_+ {1}_{ \exists \beta' \supsetneq\beta,\, {\boldsymbol{\Theta}} \in C_{\beta'}} \Big]\, .\end{align*}

The first expectation is then equal to

\begin{equation*}{\mathbb{E}} \Big[ \Big(\min_{j \in \beta^c} \Theta_{\beta, \, j, \, +}^\alpha - \Big(\sum_{k \in \beta} (\Theta_k)_+\Big)^\alpha\Big)_+ {1}_{\exists l \in \beta, \, \Theta_l =0} \Big]\,\end{equation*}

and thus vanishes. The second expectation is smaller than ${\mathbb{P}}({\boldsymbol{\Theta}} \in C_{\beta'} \text{ for some } \beta' \supsetneq\beta)$ , which is equal to zero. Indeed, if ${\mathbb{P}}({\boldsymbol{\Theta}} \in C_{\beta'} \text{ for some } \beta' \supsetneq\beta) > 0$ , then by Equation (3.4), we also have ${\mathbb{P}}({\textbf{Z}} \in C_{\beta'} \text{ for some } \beta' \supsetneq\beta) >0$ , which contradicts the maximality of $\beta$ for ${\textbf{Z}}$ . All in all this proves that $E_2 = 0$ .

Going back to Equation (C.23), we have proved that

\begin{equation*}{\mathbb{P}}({\textbf{Z}} \in C_\beta)= E_1= {\mathbb{E}}\Big[1 - \max_{j \in \beta} \Theta_{\beta, \, j, \, +}^\alpha \mid {\boldsymbol{\Theta}} \in C_\beta\Big] {\mathbb{P}} ( {\boldsymbol{\Theta}} \in C_\beta )\, .\end{equation*}

By Lemma 6 we know that the expectation is positive. Hence, the assumption ${\mathbb{P}}({\textbf{Z}} \in C_\beta) > 0$ implies that ${\mathbb{P}}({\boldsymbol{\Theta}} \in C_\beta) > 0$ , which proves that $\beta$ is a maximal direction of ${\boldsymbol{\Theta}}$ .