3.1 Aggregation across Time

Before we begin with the formal discussion, we show that linear conditional means have a simple stability property. In essence, if a Lévy process has an increment with a linear conditional mean then the conditional mean of any other increment is linear as well. This property does not require bounded second moments. It suffices to assume existence of the mean only.

In the remaining part of this section, we show that, under certain assumptions, the conditional mean $\mathbb {E}(Y_{t} | X_{t})$ of the process $V_t$ becomes asymptotically linear if $t/\tau $ is sufficiently large regardless of the functional form of $\mathbb {E}(Y_{\tau } | X_{\tau })$ and we estimate the rate of convergence. The main assumptions are listed below.

Assumption 2. The following moment conditions hold:

(11) $$ \begin{align} \mathbb{E}\left((Y_{\Delta t} - X_{\Delta t}^{\mathsf{T}} \theta^*)X^{\nu}_{\Delta t}\right) &= 0, \quad \forall \nu\colon |\nu| \leq k \end{align} $$

(12) $$ \begin{align} \mathbb{E}\left(|Y_{\Delta t} - X_{\Delta t}^{\mathsf{T}}\theta^*|\left\lVert X_{\Delta t}\right\rVert_1^{r}\right) &< \infty, \end{align} $$

where $k \geq 1$ is an integer, $\theta ^*$ is provided in equation (4), $\nu $ denotes a multi-index, and $r = k + \delta $ with $\delta \in (0, 1]$ .

The role of the assumptions can be explained in the following way. Assumption 1 defines the process and guarantees existence of $\theta ^*$ while Assumption 2 ensures boundedness of the functional $\boldsymbol {b}_r$ defined in equation (8).Footnote ⁵

Concerning the moment conditions in equation (11), note that $V_{\Delta t}$ is assumed to have a mean of zero in Assumption 1, which guarantees equation (11) holds for $k = 0$ . From the definition of $\theta ^*$ in equation (4), it follows directly that $\mathbb {E}[(Y_{\Delta t} - X_{\Delta t}^{\mathsf {T}}\theta ^*)X_{\Delta t}] = 0$ and, therefore, equation (11) holds automatically for $k = 1$ as well. Thus, the moment conditions are needed only for $k \geq 2$ .

The condition in equation (12) looks complicated but a simpler upper bound is easy to obtain. Indeed, the next inequality holds

(13) $$ \begin{align} \mathbb{E}\left(|Y_{\Delta t} - X_{\Delta t}^{\mathsf{T}}\theta^*|\left\lVert X_{\Delta t}\right\rVert_1^{r}\right) \leq \max(1, \left\lVert\theta^*\right\rVert_{\infty})\mathbb{E}\left\lVert V_{\Delta t}\right\rVert_1^{r+1}. \end{align} $$

An advantage is that we can isolate a simple moment condition involving the vector $V_{\Delta t}$ separately from $\theta ^*$ .

The main result is provided in the following theorem.

Theorem 2. Under Assumptions 1 and 2,

(14) $$ \begin{align} \boldsymbol{\varsigma}(n^{-1/2}V_{n\Delta t}) \leq C_{r}n^{1-\frac{r + 1}{2}}\boldsymbol{b}_{r}(V_{\Delta t}), \end{align} $$

where $r \geq 1$ , $C_r = \sup _u\left \lVert u\right \rVert _1^r\psi (u)$ . Finally, $\boldsymbol {b}_{r}(V_{\Delta t})$ can be bounded in the following way:

(15) $$ \begin{align} \boldsymbol{b}_{r}(V_{\Delta t}) \leq A_\delta \frac{\Gamma(\delta + 1)}{\Gamma(r + 1)}\mathbb{E}\left(|Y_{\Delta t} - X_{\Delta t}^{\mathsf{T}}\theta^*|\left\lVert X_{\Delta t}\right\rVert_1^{r}\right), \end{align} $$

in which $r = k + \delta $ , $k \geq 1$ is an integer, and $\delta \in (0, 1]$ , $A_{\delta } = \sup _{y \in \mathbb {R}}|y|^{-\delta }|e^{iy} - 1|$ is an absolute constant, and $\Gamma (x) = \int _0^\infty u^{x-1}e^{-u}du$ , $\operatorname {Re}(x)> 0$ , denotes the gamma function.

The constant $C_r$ depends entirely on the choice of the function $\psi $ and can be computed explicitly in some instances. For example, we prove in Lemma A.4 that if $\psi $ is chosen to be the ch.f. of a d-dimensional Gaussian distribution with standardized independent components, $\psi (u) = \exp (-u^{\mathsf {T}} u/2)$ , then $C_r = e^{-\frac {r}{2}}(rd)^{\frac {r}{2}}$ . It is important to emphasize that the rate of convergence does not depend on the choice of $\psi $ .

Next, we consider an important special case. If we set $r = 1 + \delta $ with $\delta \in (0, 1]$ , then, as explained previously, the moment conditions in equation (11) are satisfied automatically. We obtain the following corollary.

Corollary 1. If Assumption 1 holds together with the moment condition in equation (12) with $r = 1 + \delta $ , $\delta \in (0, 1]$ , then

$$ \begin{align*} \boldsymbol{\varsigma}(n^{-1/2} V_{n\Delta t}) &\leq n^{-\delta/2}C_{1+\delta}\boldsymbol{b}_{1+\delta}(V_{\Delta t}). \end{align*} $$

The corollary illustrates that the conditions in Assumption 2 determine the rate of decay in Theorem 2. Clearly, the rate of decay increases with r. At the limit, if $\mathbb {E}\left \lVert V_{\Delta t}\right \rVert ^{r} < \infty $ for all r, then, by (13), the condition in equation (12) holds for all r. In addition, if the conditions in equation (11) hold for all k, then for any analytic function $h\colon \mathbb {R}^d \rightarrow \mathbb {C,}$ we have

$$ \begin{align*} \mathbb{E}[(Y_{\Delta t} - X_{\Delta t}^{\mathsf{T}}\theta^*)h(X_{\Delta t})] &= \mathbb{E}\left[(Y_{\Delta t} - X_{\Delta t}^{\mathsf{T}}\theta^*)\sum_{|\nu| = 0}^{\infty}\frac{\partial^\nu h(0)}{\nu!}X_{\Delta t}^{\nu}\right]\\ &= \sum_{|\nu| = 0}^{\infty}\frac{\partial^\nu h(0)}{\nu!}\mathbb{E}\left[(Y_{\Delta t} - X_{\Delta t}^{\mathsf{T}}\theta^*)X_{\Delta t}^{\nu}\right] \\ &= 0. \end{align*} $$

The family of functions $\{h_u(x) = e^{iu^{\mathsf {T}} x}, u\in \mathbb {R}^d\}$ belongs to that class and, therefore, $\mathbb {E}[(Y_{\Delta t} - X_{\Delta t}^{\mathsf {T}}\theta ^*)h_u(X_{\Delta t})] = \mathbb {E}[(Y_{\Delta t} - X_{\Delta t}^{\mathsf {T}}\theta ^*)e^{iu^{\mathsf {T}} X_{\Delta t}}] = 0$ for all $u\in \mathbb {R}^d$ , which means that $\mathbb {E}[Y_{\Delta t}|X_{\Delta t}] = X_{\Delta t}^{\mathsf {T}}\theta ^*$ almost surely. By Lemma A.5, $\boldsymbol {b}_r(V_{n\Delta t}) = 0$ and the upper bound in equation (14) equals zero for any n.

The moment conditions in equation (11) permit an econometric interpretation. It is straightforward to notice that they represent first-order conditions of a least-squares problem. Consider, for example, the case $k = 2$ . If we add the components of $X_{\Delta t}$ raised to the second power and all mixed products linearly, we obtain the following least-squares problem:

$$\begin{align*}(\tilde{\theta}, \tilde{\beta}) = \arg\min_{(\theta, \beta)} \mathbb{E}\left(Y_{\Delta t} - X_{\Delta t}^{\mathsf{T}}\theta - \sum_{|\nu|=2}X_{\Delta t}^{\nu}\beta_{\nu}\right)^2, \end{align*}$$

where $\beta $ is a vector containing all the $\beta _\nu $ parameters. If the optimal solution is $(\tilde {\theta }, \tilde {\beta }) = (\theta ^*, 0)$ , then the moment conditions in equation (11) hold with $k = 2$ . Clearly, $\tilde {\beta } = 0$ does not necessarily imply a linear conditional mean but according to Theorem 2 it does imply a faster rate of convergence if the corresponding condition in equation (12) holds as well. The same analysis can be repeated when $k> 2$ .

The rate of decay in Theorem 2 is also affected by the choice of $\delta $ . The conditions in equations (12) and (13) indicate that $\delta $ is determined by the tail behavior of $V_{\Delta t}$ . For instance, suppose that the conditions in equation (11) hold for some fixed value of k and $\mathbb {E}\|V_{\Delta t}\|_{1}^{k + \delta + 1} < \infty $ for all $0 < \delta < \delta ^* \leq 1$ but $\mathbb {E}\|V_{\Delta t}\|_{1}^{k + \delta ^* + 1}$ is unbounded. This is possible for distributions with a Pareto-type tail, see Section 5.1.2 for an example. In this case, $\delta $ can be chosen to be close but below $\delta ^*$ bearing in mind a potential trade-off—as $\delta $ approaches $\delta ^*$ the rate of decay improves but the value of the scaling constant $\boldsymbol {b}_r(V_{\Delta t})$ may deteriorate. If the moment $\mathbb {E}\|V_{\Delta t}\|_{1}^{k + 2}$ exists, then we can set $\delta = 1$ . For example, if $\mathbb {E}\|V_{\Delta t}\|_{1}^{3} < \infty ,$ then Corollary 1 holds with $\delta = 1$ .

Another special case can be constructed by imposing additional properties on the random vector $V_{\Delta t}$ . Suppose that $V_{\Delta t}$ is symmetric, $V_{\Delta t} \overset {\scriptscriptstyle d}{=} -V_{\Delta t}$ . It is easy to see that the conditions in equation (11) hold for any even value of $|\nu |$ . As a result, the conditional mean converges to a linear function faster for symmetric distributions.

Corollary 2. If $V_{\Delta t} \overset {\scriptscriptstyle d}{=} -V_{\Delta t}$ and Assumption 1 holds together with the moment condition in equation (12) with $r = 2 + \delta $ , $\delta \in (0, 1]$ , then

$$\begin{align*}\boldsymbol{\varsigma}(n^{-1/2} V_{n\Delta t}) \leq n^{-\frac{1 + \delta}{2}}C_{2+\delta}\boldsymbol{b}_{2+\delta}(V_{\Delta t}). \end{align*}$$

Modern CLT rate-of-convergence results for metrics with uniform structure include moment conditions similar to the ones in Assumption 2, see, for example, Rachev et al. (Reference Rachev, Klebanov, Stoyanov and Fabozzi2013, Thms 15.4.2 and 15.4.3 and Remark 15.4.4) and Section 4.1 for a more detailed discussion. In terms of our notation, these moment conditions take the following form:

(16) $$ \begin{align} \mathbb{E}(V_{\Delta t}^\gamma - Z^\gamma) &= 0,\quad \forall \gamma\colon |\gamma| \leq k \end{align} $$

(17) $$ \begin{align} {\boldsymbol{v}}_{r+1}(V_{\Delta t}, Z) &= \int_{\mathbb{R}^{d+1}}\left\lVert\xi\right\rVert_1^{r+1}|F - \Phi|(d\xi) < \infty, \end{align} $$

where $r = k + \delta $ , $\delta \in (0, 1]$ , $\gamma $ is a multi-index, Z denotes a $(d+1)$ -dimensional Gaussian vector, and F and $\Phi $ denote the distribution functions of $V_{\Delta t}$ and Z, respectively. The functional ${\boldsymbol {v}}_{s}$ is known as an absolute pseudomoment of order s and its role is illustrated in Section 4.1. Even though the left side of equation (11) for any fixed $\nu $ is a linear combination of some of the moments $\mathbb {E} [V_{\Delta t}^{\nu + 1}]$ , one key difference from equation (16) is that the conditions in Assumption 2 do not involve moments of the marginals of $V_{\Delta t}$ on a stand-alone basis. For this reason, the conditions in Assumption 2 are weaker than those in equation (16) and the rate of decay in Theorem 2 can be faster than the corresponding rate of convergence in the CLT implied by the conditions in equations (16) and (17).

A somewhat trivial illustration of this observation can be constructed in the following way. If we choose any multivariate law for $V_{\Delta t}$ with linear conditional expectations, then, by Theorem 1 and Lemma A.1, we have $\boldsymbol {\varsigma }(V_{n\Delta t}) = 0$ for all $n\geq 1$ regardless of the marginal distributions of $V_{\Delta t}$ . At the same time, the conditions in equation (16) may not hold for any $k \geq 3$ and the rate of convergence in the CLT for the chosen multivariate law may be as slow as the classical rate of $n^{-1/2}$ . Section 5.2 provides a non-trivial numerical example of this phenomenon.

3.2 Cross-Sectional Aggregation Combined with Temporal Aggregation

The simulation study by Granger and Lee (Reference Granger and Lee1999) concludes that if there are prominent factors with a nonlinear impact at micro-level, the nonlinear effects may persist in cross-sectionally aggregated data. In some applications, however, the aggregation process is applied jointly cross-sectionally and in the time dimension. In this section, we focus on the joint impact.

In finance, for example, portfolios are formed from individual stocks and portfolio returns can be viewed as cross-sectional weighted averages of stock returns while return frequency reflects temporal aggregation. In fact, factor models are commonly applied to portfolios rather than individual basic risky assets. In this section, we change the setting to accommodate linear combinations of Y variables interpreted as portfolios. We demonstrate that under fairly general conditions linear models can become more adequate for portfolios faster with n than for individual stocks when the data are temporally aggregated.

Consider a multivariate process $V_t = (Y_t, X_t) \in \mathbb {R}^m\times \mathbb {R}^d$ , where m denotes the number of basic risky assets and d denotes the number of factors. The linear combination $Y_{p,t} = \sum _{i = 1}^mw_iY_{i,t}$ denotes a portfolio of the basic risky assets, where $w_i \geq 0$ stand for portfolio weights. Even though we focus on long-only portfolios, a similar analysis holds in the general case. Finally, denote by ${\mathbf {T}}_jV_t = (Y_{j,t}, X_t)$ an operator that selects the j-th component of $Y_t$ together with all factors and let ${\mathbf {T}}_pV_t = (Y_{p,t}, X_t)$ be an operator that constructs the portfolio and selects all factors.

The two assumptions can be adapted in the following way. Assumption 1 holds for the whole vector $V_t$ while Assumption 2 is modified to hold only for a specific portfolio.

Assumption 2*. The following moment conditions hold for ${\mathbf {T}}_pV_{\Delta t} = (Y_{p,\Delta t}, X_{\Delta t})$ :

(18) $$ \begin{align} \mathbb{E}\left((Y_{p,\Delta t} - X_{\Delta t}^{\mathsf{T}} \theta^*_p)X^{\nu}_{\Delta t}\right) &= 0, \quad \forall \nu\colon |\nu| \leq k_p \end{align} $$

(19) $$ \begin{align} \mathbb{E}\left(|Y_{p,\Delta t} - X_{\Delta t}^{\mathsf{T}}\theta^*_p|\left\lVert X_{\Delta t}\right\rVert_1^{r_p}\right) &< \infty, \end{align} $$

where $Y_{p,t} = \sum _{i = 1}^mw_iY_{i,t}$ , $k_p \geq 1$ is an integer, $\nu $ denotes a multi-index, $r_p = k_p + \delta $ with $\delta \in (0, 1]$ , and $\theta ^*_p$ is provided by

$$ \begin{align*} \theta^*_p = (\mathbb{E}[X_{\Delta t}X_{\Delta t}^{\mathsf{T}}])^{-1}\mathbb{E}[Y_{p,\Delta t}X_{\Delta t}]. \end{align*} $$

The next corollary follows directly from Theorem 2.

Corollary 3. Under Assumptions 1 and 2*,

(20) $$ \begin{align} \boldsymbol{\varsigma}(n^{-1/2}{\mathbf{T}}_pV_{n\Delta t}) \leq C_{r_p}n^{1-\frac{r_p + 1}{2}}\boldsymbol{b}_{r_p}({\mathbf{T}}_pV_{\Delta t}), \end{align} $$

where $r_p \geq 1$ .

Cross-sectional aggregation can improve the upper bound in (20) relative to the upper bounds of the individual assets when Corollary 3 is applied independently to ${\mathbf {T}}_jV_t$ . For example, let us assume the worst-case scenario. If $\mathbb {E}\left \lVert V_{\Delta t}\right \rVert _1^{2+\delta } < \infty $ , then (19) holds for any portfolio and any individual risky asset, respectively. As illustrated previously, the condition in (18) holds automatically for $k_p = 1$ for any portfolio as well. Therefore, the inequality in (20) takes the form

$$ \begin{align*} \boldsymbol{\varsigma}(n^{-1/2}{\mathbf{T}}_pV_{n\Delta t}) &\leq C_{1+\delta}n^{-\frac{\delta}{2}}\boldsymbol{b}_{1+\delta}({\mathbf{T}}_pV_{\Delta t})\\ &\leq C_{1+\delta}n^{-\frac{\delta}{2}}\sum_{j = 1}^mw_j\boldsymbol{b}_{1+\delta}({\mathbf{T}}_jV_{\Delta t}), \end{align*} $$

where the second inequality follows easily from $\theta _p = \sum _{i = 1}^mw_i\theta _i^*$ and the definition of $\boldsymbol {b}_r$ . Even though in this case the rate of convergence is the same regardless of the weights, the corresponding upper bound for the portfolio is smaller than the average of the individual risky assets’ upper bounds.

Next, suppose $\mathbb {E}\left \lVert V_{\Delta t}\right \rVert _1^{3+\delta } < \infty $ . Also, assume the condition in equation (18) does not hold for $\nu \colon |\nu |> 1$ for any of the individual assets. The condition in (18) for any portfolio can be expressed through the weights and the individual assets in the following way:

$$ \begin{align*} \mathbb{E}\left((Y_{p,\Delta t} - X_{\Delta t}^{\mathsf{T}} \theta^*_p)X^{\nu}_{\Delta t}\right) &= \mathbb{E}\left[\left(\sum_{j = 1}^mw_jY_{j,\Delta t} - X_{\Delta t}^{\mathsf{T}} \left(\sum_{j = 1}^mw_j\theta^*_j\right)\right)X^{\nu}_{\Delta t}\right]\\ &= \sum_{j = 1}^mw_j\mathbb{E}\left((Y_{j,\Delta t} - X_{\Delta t}^{\mathsf{T}} \theta^*_j)X^{\nu}_{\Delta t}\right). \end{align*} $$

Thus, the condition in (18) for $k_p = 2$ can be written as

(21) $$ \begin{align} \sum_{j = 1}^mw_j\mathbb{E}\left((Y_{j,\Delta t} - X_{\Delta t}^{\mathsf{T}} \theta^*_j)X^{\nu}_{\Delta t}\right) = 0,\qquad \forall \nu\colon |\nu| = 2, \end{align} $$

because it holds for $\nu \colon |\nu | = 1$ by assumption. The number of equations in (21) equals $d + d(d-1)/2$ and this number depends only on the dimension d of the vector X. All of the equations are linear with respect to the portfolio weights. Therefore, if

(22) $$ \begin{align} d + \frac{d(d - 1)}{2}< m, \end{align} $$

then there exists a portfolio (possibly a long-short one) for which (21) holds. As a consequence, for that portfolio $r_p = 2+\delta $ and the inequality in (20) becomes

$$\begin{align*}\boldsymbol{\varsigma}(n^{-1/2}{\mathbf{T}}_pV_{n\Delta t}) \leq C_{2+\delta}n^{-\frac{\delta + 1}{2}}\boldsymbol{b}_{2+\delta}({\mathbf{T}}_pV_{\Delta t}). \end{align*}$$

As a result, the rate of decay $(\delta + 1)/2$ is strictly faster than the rate $\delta /2$ of the individual risky assets. Section 5.1.3 includes a simple illustration with the hyperbolic skewed-t distribution.

In the practice of finance, factor models are usually of low dimension while the cross section, depending on which universe is considered, typically contains a large number of stocks. Thus, the left side of the inequality in equation (22) is in practice much smaller than m. For example, $m = 1,000$ for the large cap Russell 1,000 universe, while for a ten-factor model, the left side equals 55.Footnote ⁶ In the academic literature, very popular factor models include the three-factor model of Fama and French (Reference Fama and French1992), the four-factor model of Carhart (Reference Carhart1997), and the five-factor model of Fama and French (Reference Fama and French2015). Even though there is a debate about the dimension of the factor space, it is generally believed to be low (Feng et al., Reference Feng, Giglio and Xiu2020).

4.1 A Connection with the Generalized CLT

It is important to note that in the infinite variance case, the limit law may have nonlinear conditional means, see Samorodnitsky and Taqqu (Reference Samorodnitsky and Taqqu1994, Chap. 5) for a complete characterization of the bivariate case. Therefore, if no vector of parameters solves equations (24) and (25), then $\boldsymbol {\varsigma }(n^{-1/\alpha }V_{n\Delta t})$ does not decay. Next, we discuss this situation in more detail and we clarify a connection with a classical rate-of-convergence result in the GCLT setting. To simplify the expressions, we denote the expectation appearing in $\boldsymbol {\varsigma }(n^{-1/\alpha }V_{n\Delta t})$ and the corresponding expectation of the limit by

$$ \begin{align*} \varpi_{\alpha, n}^\theta(u) &= \mathbb{E}\left[\frac{1}{n^{1/\alpha}}(Y_{n\Delta t} - X_{n\Delta t}^{\mathsf{T}}\theta)e^{iu^{\mathsf{T}} \frac{1}{n^{1/\alpha}}X_{n\Delta t}}\right]\\ \varpi_{\alpha}^\theta(u) &= \mathbb{E}(Z_{y,\Delta t} - Z_{x,\Delta t}^{\mathsf{T}}\theta)e^{iu^{\mathsf{T}} Z_{x,\Delta t}}, \end{align*} $$

where $\theta $ is the same in both expressions.

The result in this section involves the following weighted metric between the ch.f. of two random vectors U and W,

(32) $$ \begin{align} {\boldsymbol{\chi}}_s(U,W) = \sup_u\left\lVert u\right\rVert_1^{-s}|\mathbb{E} e^{iu^{\mathsf{T}} U} - \mathbb{E} e^{iu^{\mathsf{T}} W}|, \end{align} $$

where $s \geq 0$ . A classical result in the rate-of-convergence problem for GCLT restated in our notation shows that if the next moment conditions hold,

(33) $$ \begin{align} \int_{\mathbb{R}^{d+1}}z^\gamma d(F - G) = 0, \ \forall \gamma\colon |\gamma| \leq \ell \quad \mathrm{and}\quad {\boldsymbol{v}}_{s}(V_{\Delta t}, Z_{\Delta t})< \infty, \end{align} $$

where $s = \ell + \delta $ , $\ell \geq 1$ is an integer, $\delta \in (0, 1]$ , the absolute pseudomoment ${\boldsymbol {v}}_s$ is provided in equation (17), $Z_{\Delta t}$ has an $\alpha $ -stable distribution, and $\gamma $ is a multi-index, then

(34) $$ \begin{align} {\boldsymbol{\chi}}_s\left(n^{-1/\alpha}V_{n\Delta t},\,n^{-1/\alpha}Z_{n\Delta t}\right) \leq n^{1 - s/\alpha}{\boldsymbol{\chi}}_s(V_{\Delta t}, Z_{\Delta t}), \end{align} $$

in which $s> \alpha $ and the right side is finite (Rachev et al., Reference Rachev, Klebanov, Stoyanov and Fabozzi2013, Chap. 15). The metric ${\boldsymbol {\chi }}_s$ satisfies a set of properties which define it as an ideal probability metric of order s and the inequality in equation (34) is a direct consequence of them.Footnote ⁷ The conditions in equations (16) and (17) are the same as the ones in equation (33) when $Z_{\Delta t}$ has a multivariate Gaussian distribution and the upper bound in equation (34) holds for the CLT with $\alpha = 2$ .

The connection between the GCLT and Theorem 3 is provided below. Basically, the moment conditions in equation (33) replace Assumption 2^† and the limiting distribution is not assumed to have a linear conditional mean; that is, the part of Assumption 1^† involving the existence of $\theta _\alpha ^*$ is ignored.

Theorem 4. Suppose the random vector $V_{\Delta t} = (Y_{\Delta t}, X_{\Delta t}) \in \mathbb {R}\times \mathbb {R}^d$ is an increment of a Lévy process with $\mathbb {E}(V_{\Delta t}) = 0$ with distribution function F and $Z_{\Delta t} = (Z_{y,\Delta t}, Z_{x,\Delta t}) \in \mathbb {R}\times \mathbb {R}^d$ is an increment of a strictly $\alpha $ -stable process, $1 < \alpha \leq 2$ , with distribution function G. Also, suppose that the moment conditions in equation (33) hold with $s = k + \delta $ , $k \geq 2$ , $\delta \in (0, 1]$ . Then, the next inequality holds for any bounded $\theta $ ,

(35) $$ \begin{align} \begin{aligned} \sup_u\left\{\psi(u)\left|\varpi_{\alpha, n}^\theta(u) - \varpi_{\alpha}^\theta(u)\right|\right\}&\leq C_{s-1}n^{1 - \frac{s}{\alpha}}\boldsymbol{b}_{s-1}^\theta(V_{\Delta t}, Z_{\Delta t}) \\ &\quad + \alpha C_{s - 1 +\alpha}D_\alpha n^{1 - \frac{s}{\alpha}}{\boldsymbol{\chi}}_s(X_{\Delta t}, Z_{x,\Delta t}), \end{aligned} \end{align} $$

where $\psi (u)> 0$ is the ch.f. of a symmetric random vector with a sufficiently smooth density, $C_p = \sup _u\left \lVert u\right \rVert _1^p\psi (u)$ , $D_\alpha = \sup _{t\in B}\left |\omega _{\alpha ,\Lambda }(t)\right |$ , in which $B = \{z\in \mathbb {R}^d\colon \left \lVert z\right \rVert _1=1\}$ and $\omega _{\alpha ,\Lambda }(t)$ is provided in Lemma A.9.

To illustrate the connection with Theorem 3, represent $z = (y, x_1, \ldots , x_d)$ and suppose that the conditions in equation (33) hold for some $\ell $ . Then, we have

(36) $$ \begin{align} \begin{aligned} \int_{\mathbb{R}^{d+1}}yx^\nu d(F - G) = 0\quad \text{and}\quad \int_{\mathbb{R}^{d+1}}x_jx^\nu d(F - G) = 0 \end{aligned} \end{align} $$

for all multi-indices $\nu \colon |\nu | = \ell -1$ and $j = 1,\ldots , d$ . It is easy to see that any linear combination of the integrals on the left side of (36) also vanishes and, therefore, the condition in equation (26) in Assumption 2^† holds for any choice of $\theta $ when $k = \ell - 1$ . Finally, the boundedness of the absolute pseudomoment ${\boldsymbol {v}}_s$ assumed in (33) combined with the inequality in (29) ensures that the moment condition in (27) in Assumption 2^† holds with $r = s - 1$ as well. As a consequence, Lemma A.8 shows that under these assumptions the functional $\boldsymbol {b}_r^\theta (V_{\Delta t}, Z_{\Delta t})$ exists for any bounded $\theta $ .Footnote ⁸

Since all quantities in the upper bound in (35) are finite, its right side provides a rate of convergence of $\varpi _{\alpha , n}^\theta $ to $\varpi _{\alpha }^\theta $ under the weighted uniform distance and, clearly, $\varpi _{\alpha }^\theta $ may not vanish for all values of u. However, if the corresponding conditional mean of the limit law is linear, then $\varpi _{\alpha }^{\theta }(u) = 0$ for all $u\in \mathbb {R}^d$ for some $\theta = \theta _\alpha ^*$ . In this situation, the constant $D_\alpha $ vanishes as well because in this case there exists $\theta _\alpha ^*$ satisfying equations (24) and (25) which, in turn, ensure $\omega _{\alpha ,\Lambda }(t) = 0$ for all $t \in \mathbb {R}^d$ .Footnote ⁹ As a result, the left side of (35) turns into $\boldsymbol {\varsigma }(n^{-1/\alpha }V_{n\Delta t})$ and the second term on the right side vanishes, which makes the right side of (35) the same as the upper bound in equation (30) in Theorem 3. Even though (30) can be viewed as a special case of equation (35), in practice, Theorem 3 is proved under weaker conditions.

5.1 Hyperbolic Skewed-t Examples

The hyperbolic skewed-t process is a special case of the generalized hyperbolic process which has received a lot of attention in the finance literature (Eberlein, Reference Eberlein, Barndorff-Nielsen, Mikosch and Resnick2001; McNeil, Frey, and Embrechts, Reference McNeil, Frey and Embrechts2005; Bianchi et al., Reference Bianchi, Stoyanov, Tassinari, Fabozzi and Focardi2019). Suppose that the vector $V_1 \in \mathbb {R}^{m + d}$ has a hyperbolic skewed-t distribution denoted by $V_1 \in \mathrm {HSt}_{\lambda }(\mu , \beta , \Sigma , \chi )$ and defined by the following ch.f.:

$$ \begin{align*} \phi_1(z) = \frac{2^{1 + \lambda}e^{iz^{\mathsf{T}}\mu}}{\Gamma(-\lambda)}\left(-2\chi\left(iz^{\mathsf{T}}\beta - \frac{1}{2}z^{\mathsf{T}}\Sigma z\right)\right)^{-\lambda/2}K_\lambda\left(\sqrt{-2\chi\left(iz^{\mathsf{T}}\beta - \frac{1}{2}z^{\mathsf{T}}\Sigma z\right)}\right), \end{align*} $$

in which we set $z = (v, u) \in \mathbb {R}^{m + d}$ and $K_\lambda (x) = \frac {1}{2}\int _{0}^{\infty }t^{\lambda - 1}e^{-x(t+t^{-1})/2}dt$ , $x> 0, $ denotes the Bessel function of the second kind. The parameters have the following interpretations: $\mu \in \mathbb {R}^{m + d}$ is a location parameter, $\beta \in \mathbb {R}^{m + d}$ is a skewness parameter, $\Sigma $ is an $(m + d)\times (m + d)$ symmetric matrix, $\chi> 0$ acts like a scaling parameter, and $\lambda < 0$ is a shape parameter.

Most of the intuition behind the interpretation of the parameters follows from the normal location-scale mixture representation:

(39) $$ \begin{align} V_1 = \mu + \beta W + \sqrt{W}Z, \end{align} $$

where W has an inverse gamma distribution $\mathrm {IG}(-\lambda , \chi /2)$ and $Z \in \mathcal {N}(0, \Sigma )$ has a multivariate Gaussian distribution with covariance matrix $\Sigma $ . From this representation, it is easy to see that linear transformations belong to the same class. Suppose that $\mathbf {A}$ is a $q\times (m + d)$ matrix. Then, $\mathbf {A} V_1 \in \mathrm {HSt}_\lambda (\mathbf {A}\mu , \mathbf {A}\beta , \mathbf {A}\Sigma \mathbf {A}^{\mathsf {T}}, \chi )$ .

Following Section 3.2, partition the random vector $V_1$ in the following manner $V_1 = (Y_1, X_1) \in \mathbb {R}^m\times \mathbb {R}^d$ . The parameters of the distribution are partitioned accordingly

$$ \begin{align*} \mu = \begin{pmatrix} \mu_Y \\ \mu_X \end{pmatrix},\qquad \beta = \begin{pmatrix} \beta_Y \\ \beta_X \end{pmatrix},\qquad \Sigma = \begin{pmatrix} \Sigma_{YY} & \Sigma_{YX} \\ \Sigma_{XY} & \Sigma_{XX} \end{pmatrix}. \end{align*} $$

Jamalizadeh and Balakrishnan (Reference Jamalizadeh and Balakrishnan2019) provide an expression for the conditional mean of a general normal location-scale mixture which takes the following form:

(40) $$ \begin{align} \mathbb{E}(Y_1 | X_1 = x) = \mu_Y + \Sigma_{YX}\Sigma_{XX}^{-1}(x - \mu_X) + (\beta_{Y} - \Sigma_{YX}\Sigma_{XX}^{-1}\beta_{X})\mathbb{E} W_x, \end{align} $$

where $W_x$ denotes the conditional distribution $W|(X_1 = x)$ . As a consequence, the conditional mean is linear if and only if $\beta _{Y} = \Sigma _{YX}\Sigma _{XX}^{-1}\beta _{X}$ . Even though the expectation $\mathbb {E} W_x$ can be computed explicitly for the hyperbolic skewed-t distribution, we do not work with these expressions because the class is not closed with respect to the convolution operation. However, the ch.f. of the normalized sum $\widetilde {V}_n$ is easy to write down and the conditional mean can be computed numerically using equation (38).

The hyperbolic skewed-t has both finite and infinite variance representatives. The tail behavior of the distribution can be summarized in the following way. If we take a linear combination $w^{\mathsf {T}} V_1$ with $w^{\mathsf {T}} \beta> 0$ , then the asymptotic decay of the density is given by

$$ \begin{align*} f_{w^{\mathsf{T}} V_1}(y) &\sim |y|^{\lambda - 1}e^{-2w^{\mathsf{T}} \beta y},\qquad y\rightarrow\infty\\ f_{w^{\mathsf{T}} V_1}(y) &\sim |y|^{\lambda - 1},\qquad y\rightarrow -\infty. \end{align*} $$

As a consequence, $V_1$ has a finite expectation if $\lambda < -1$ and a finite variance if $\lambda < -2$ . The asymptotic property of the tail also suggests the correct exponent $\alpha $ in the normalization in (37) is $\alpha = \min (-\lambda , 2)$ .

5.1.1 Convergence to a Linear Function

Our first example is an illustration of the convergence result in Section 3. We assume $m = 1$ , $d = 1$ and that $V_1$ has a finite variance and a nonlinear conditional mean. In terms of parameters, we pick $\lambda = -5$ , $\chi = 1$ ,

$$\begin{align*}\beta = \begin{pmatrix} -2 \\ 2 \end{pmatrix},\qquad \mu = \frac{\chi}{2(\lambda + 1)}\beta, \qquad \text{and}\qquad \Sigma = \begin{pmatrix} 1 & 0.5 \\ 0.5 & 1 \end{pmatrix}. \end{align*}$$

The value for $\mu $ ensures that the distribution has a mean of zero, the remaining values are chosen for convenience. The shape of the conditional mean as function of the number of summands n is provided in Figure 1. The x-axis covers the range from the 1%-quantile to the 99%-quantile of the distribution of $X_1$ . In line with the developed theory, the conditional mean converges to a linear function.

Figure 1

The conditional mean of the normalized sum as a function of the number of summands n when $V_1$ has a finite variance and a nonlinear conditional mean. $V_1 = (Y_1, X_1)$ follows a hyperbolic skewed-t with parameters $\lambda = -5$ and $\beta = (-2, 2)$ . The conditional mean converges to a linear function.

5.1.2 Rate of Convergence for Mixture Models

An interesting question is if there exist parameter values such that the conditions in equation (11) hold with $k \geq 2$ . Consider again the case $m = 1$ , $d = 1$ and assume that all parameters are fixed except for $\beta _Y$ . Given that $\mathbb {E}(V_1) = 0$ , the stochastic representation in (39) can be restated equivalently as

(41) $$ \begin{align} \begin{aligned} Y_1 &= \beta_Y(W - \mathbb{E} W) + \sqrt{W}Z_Y \\ X_1 &= \beta_X(W - \mathbb{E} W) + \sqrt{W}Z_X. \end{aligned} \end{align} $$

The third-order condition in equation (11) corresponding to $k = 2$ takes the form $\mathbb {E}(Y_1X_1^2) - \theta ^*\mathbb {E} X_1^3 = 0$ . This is a linear equation in $\beta _Y$ because $\theta ^* = \mathbb {E}(Y_1X_1)/\mathbb {E}(X_1^2)$ is also linear in $\beta _Y$ and the only solution is $\beta _Y = \sigma _{YX}\beta _X/\sigma _{XX}$ found by direct verification. According to equation (40), however, this choice of $\beta _Y$ turns the conditional mean into a linear function. As a consequence, in this case, the third-order moment condition in equation (11) guarantees linearity of the conditional mean.

The argument presented above is general and the same conclusion holds for any normal location-scale mixture for which the corresponding moments exist. Corollary 1, therefore, captures the rate of convergence for location-scale mixtures with finite variance. As far as $\delta $ is concerned, if the third moments of the marginals are finite, $\lambda < -3$ , then $\delta = 1$ can be used. If the third moments are unbounded, then the upper limit on $\delta $ is determined by the tail behavior of the marginals. For instance, in the case of the hyperbolic skewed-t, if we require $-3 \leq \lambda < -2$ to guarantee existence of second moments when the third moments are unbounded, the corresponding rate of decay is provided in Corollary 1 with $0 < \delta < \min (-\lambda - 2, 1)$ .

A way to engineer a model more flexible than the classical normal location-scale mixture is to change the tail behavior of the components in equation (41). Consider the next modification,

(42) $$ \begin{align} \begin{aligned} Y_1 &= \beta_Y(h(W) - \mathbb{E} h(W)) + \sqrt{h(W)}Z_Y \\ X_1 &= \beta_X(W - \mathbb{E} W) + \sqrt{W}Z_X, \end{aligned} \end{align} $$

where $h(x) = G^{-1}(F(x))$ in which F is the c.d.f. of W and G is the c.d.f. of a positive random variable. This construction is similar to the meta-Student-t distribution featuring different degrees of freedom for the marginals and the meta-elliptical distribution (Fang, Fang, and Kotz, Reference Fang, Fang and Kotz2002; Demarta and McNeil, Reference Demarta and McNeil2005). It is easy to see from the definition in (42) that the conditional mean is nonlinear even when $\beta _X = \beta _Y = 0$ ,

(43) $$ \begin{align} \mathbb{E}(Y_1|X_1 = x) = x\frac{\sigma_{YX}}{\sigma_{XX}}\mathbb{E}\left(\frac{h(W_x)}{W_x}\right)^{1/2}, \end{align} $$

where $W_x$ denotes the conditional distribution $W|(X_1 = x)$ . At the same time, the multivariate law is symmetric when $\beta _X = \beta _Y = 0$ and the faster rate of convergence in Corollary 2 holds.

In a similar fashion, the equation $\mathbb {E}(Y_1X_1^2) - \theta ^*\mathbb {E} X_1^3 = 0$ is linear in $\beta _Y$ and can be used to determine $\beta _Y$ as a function of $\beta _X$ . The different tail behavior of the components provides more flexibility and, unlike the classical normal location-scale mixture, the moment condition for $k = 2$ can hold when the distribution is asymmetric and the conditional mean $\mathbb {E}(Y_1|X_1 = x)$ is nonlinear.

A similar argument is used in the construction in Section 5.2 in which the conditional mean is specified explicitly. A difficulty with the meta-mixture model in (43) is that the expectation on the right hand-side is not available in closed form.

5.1.3 Cross-Sectional Aggregation

The argument developed in Section 3.2 concerning portfolios can be illustrated in the following manner. Suppose $V_1 = (Y_1, X_1) \in \mathbb {R}^m\times \mathbb {R}^d$ in which the first m components are interpreted as basic assets and the last d components are interpreted as factors. The operator ${\mathbf {T}}_p$ takes the form

$$\begin{align*}{\mathbf{T}}_p = \begin{pmatrix} {\mathbf{w}}^{\mathsf{T}} & \mathbf{0}_{1\times d}\\ \mathbf{0}_{d\times m} & \mathbf{I}_{d\times d} \\ \end{pmatrix}, \end{align*}$$

where ${\mathbf {w}} = (w_1, \ldots , w_m)$ , $\sum _{j = 1}^m w_j = 1$ , denote portfolio weights. From the properties of the hyperbolic skewed-t distribution, it follows that the random vector ${\mathbf {T}}_p V_1$ has the same distribution with parameters ${\mathbf {T}}_p V_1 \in \mathrm {HSt}_\lambda ({\mathbf {T}}_p\mu , {\mathbf {T}}_p\beta , {\mathbf {T}}_p\Sigma {\mathbf {T}}_p^{\mathsf {T}}, \chi )$

$$ \begin{align*} {\mathbf{T}}_p\mu = \begin{pmatrix} {\mathbf{w}}^{\mathsf{T}}\mu_Y \\ \mu_X \end{pmatrix},\qquad \beta = \begin{pmatrix} {\mathbf{w}}^{\mathsf{T}}\beta_Y \\ \beta_X \end{pmatrix},\qquad \Sigma = \begin{pmatrix} {\mathbf{w}}^{\mathsf{T}}\Sigma_{YY}{\mathbf{w}} & {\mathbf{w}}^{\mathsf{T}}\Sigma_{YX} \\ \Sigma_{XY}{\mathbf{w}} & \Sigma_{XX} \end{pmatrix}. \end{align*} $$

The relationship between the skewness parameters that results in a linear conditional mean can be readily expressed in terms of the weights and the parameters of the distribution of $V_1$

$$\begin{align*}{\mathbf{w}}^{\mathsf{T}}(\beta_Y - \Sigma_{YX}\Sigma_{XX}^{-1}\beta_X) = 0. \end{align*}$$

Since this is a linear equation, clearly there are many portfolios with a linear conditional mean to the set of factors in $X_1$ even though all of the individual assets may have a nonlinear conditional mean to the same factor set.

5.1.4 Convergence to a Nonlinear Function

The next example concerns the case of a nonlinear limiting conditional mean in the case of infinite variance. Consider $(Y_1, X_1) \in \mathbb {R}\times \mathbb {R}$ and choose $\lambda = -1.75$ and $\beta = (0.001, 0.5)$ , which implies $\alpha = 1.75$ in equation (37). Both marginals have an infinite second moment and their limiting distribution is in the $\alpha $ -stable class. As noted in Theorem 4, in this situation, the limiting conditional mean may not be linear. Indeed, as illustrated in Figure 2, the conditional mean does not converge to a linear function.

Figure 2

The conditional mean of the normalized sum as a function of the number of summands n when $V_1$ has a nonlinear conditional mean and infinite variance. $V_1 = (Y_1, X_1)$ follows a hyperbolic skewed-t with parameters $\lambda = -1.75$ , $\beta = (0.001, 0.5)$ . The conditional mean does not converge to a linear function.

5.2 Examples of Different Rates of Convergence

Equation (11) provides the moment conditions that guarantee a given order of the convergence rate when the second moment is finite. In this section, our objective is to develop a bivariate example featuring a degree of analytic tractability in which the distribution of $X_1$ is asymmetric and remains fixed, the conditional mean is specified in closed form and its parameters can be chosen so that the moment conditions in equation (11) hold with $k \leq 2$ . Such an example is interesting also to illustrate that the moment conditions in (11) are not a trivial consequence of the CLT and the rate of convergence can be faster than the corresponding CLT rate for the same law.

Let $(Y, X) \in \mathbb {R}\times \mathbb {R}$ and assume X follows an asymmetric Laplace distribution with density

$$\begin{align*}f_{X}(x) = \begin{cases} \dfrac{\kappa_1\kappa_2}{\kappa_1 + \kappa_2}\exp\{-\kappa_1(x - \mu)\}, & x \geq \mu \\[1em] \dfrac{\kappa_1\kappa_2}{\kappa_1 + \kappa_2}\exp\{-\kappa_2(\mu - x)\}, & x < \mu, \end{cases} \end{align*}$$

where $\kappa _1,\kappa _2> 0$ . Define Y by means of the stochastic representation

(44) $$ \begin{align} Y = g(X) + \epsilon, \end{align} $$

where $\epsilon $ and X are independent, $\epsilon $ is assumed to have finite variance, and the conditional mean is defined explicitly in the following way:

(45) $$ \begin{align} g(x) = \begin{cases} (x - \mu)^{\gamma_1}, & x \geq \mu \\ -(\mu - x)^{\gamma_2}, & x < \mu, \end{cases} \end{align} $$

where $\gamma _1,\gamma _2> 0$ .

The utility of this construction is that the expectations behind $\boldsymbol {\varsigma }$ and the corresponding moment conditions can be calculated analytically in closed form. The disadvantage is that although the distribution of X is infinitely divisible, that of Y may not be and in such a case $(Y, X)$ cannot be used to construct a Lévy process. Nevertheless, all results in the article can be restated without modification for the classical CLT scheme of sums of i.i.d. random vectors and the example is relevant (see Section 5.3).

Equation (45) and the relationship $\mathbb {E}(Ye^{iuX}) = \mathbb {E}(\mathbb {E}(Y|X)e^{iuX})$ are used to arrive at the next expressions for the ch.f. of X and all relevant moments

(46) $$ \begin{align} \phi_X(u) = \mathbb{E}(e^{iuX}) &= \frac{\kappa_1\kappa_2 e^{iu\mu}}{(\kappa_1 - iu)(\kappa_2 + iu)} \end{align} $$

(47) $$ \begin{align} \mathbb{E} (Ye^{iuX}) &= \frac{\kappa_1\kappa_2e^{iu\mu}}{\kappa_1 + \kappa_2}\left[\frac{\Gamma(\gamma_1 + 1)}{(\kappa_1 - iu)^{\gamma_1 + 1}} - \frac{\Gamma(\gamma_2 + 1)}{(\kappa_2 + iu)^{\gamma_2 + 1}}\right] \end{align} $$

(48) $$ \begin{align} \mathbb{E}(Xe^{iuX}) &= \frac{\kappa_1\kappa_2e^{iu\mu}}{\kappa_1 + \kappa_2}\left[\frac{1}{(\kappa_1 - iu)^{2}} - \frac{1}{(\kappa_2 + iu)^{2}}\right] + \mu\phi_X(u). \end{align} $$

The remaining moments together with $\theta ^*$ defined in equation (4) needed to compute $\boldsymbol {\varsigma }$ and the conditions in (11) are calculated by evaluating derivatives of (46)–(48) at $u = 0$ . Both X and Y need to be centered and, therefore, the location parameter is set to $\mu = -\mathbb {E} X = -(\kappa _2 - \kappa _1)/\kappa _1\kappa _2$ . The mean of Y needs to be subtracted as well inside the expectation in (47), which will result in a new term on the right side.

We note that the actual distribution of the residual in (44) is irrelevant in these calculations. The skewed Laplace distribution has exponential tails and the variance of $g(X)$ exists. The requirement for finite variance of $\epsilon $ is needed only to make sure the variance of Y is finite.

The conditional mean $\mathbb {E}(\widetilde {Y}_n | \widetilde {X}_n = x)$ is computed in two steps. First, repeating the steps behind the derivation of (5) we find that

$$\begin{align*}\mathbb{E} (\widetilde{Y}_ne^{iu\widetilde{X}_n}) = n^{1/2}\mathbb{E} (Y e^{in^{-1/2}u X})\left[\phi_X(n^{-1/2}u)\right]^{n-1}. \end{align*}$$

Second, we use equation (38) and apply the FFT to the function $\mathbb {E} (\widetilde {Y}_ne^{iu\widetilde {X}_n})$ to evaluate numerically the conditional mean. The left plot in Figure 3 shows how the conditional mean changes as a function of n with parameter values $\kappa _1 = 2.5$ , $\kappa _2 = 1$ , and $\gamma _1 = \gamma _2 = 2$ . The distribution of X is skewed to the left and the grid for x ranges from the 1% quantile to the 99% quantile of X. With these parameter values, the third-order moment condition in (11) does not hold. The rate of convergence in Corollary 1 is expected to hold for this situation.

Figure 3

The conditional mean of the normalized sum as a function of the number of summands n with parameters $\kappa _1 = 2.5$ , $\kappa _2 = 1$ , and $\gamma _1 = \gamma _2 = 2$ (left) and $\kappa _1 = 2.5$ , $\kappa _2 = 1$ , $\gamma _1 = 5.1$ , and $\gamma _2 = 2$ (right). The plot on the right illustrates a faster rate of convergence because the choice of parameters ensures that the third-order moment condition in (11) is satisfied. The grid on the horizontal axis starts at the 1% quantile and ends at the 99% quantile of the distribution of X.

Next, we find a new value for $\gamma _1$ by numerically solving the third-order condition in (11), $\mathbb {E}[(Y - \theta ^*X)X^2] = 0$ . Conditional on the other parameters, the solution is $\gamma _1^* = 5.1> \gamma _1 = 2$ which is intuitive given that g needs to steepen to compensate for the lighter right tail of X in order for the moment condition to hold. The right plot in Figure 3 illustrates this case. Even though the conditional mean g is now more curved, the rate of convergence to the asymptotic linear case is visually faster as expected based on Theorem 2.

The actual rate of convergence in these two cases is estimated formally in the following way. The log-transform applied to both sides of (14) leads to the next asymptotic relationship for large n

(49) $$ \begin{align} \log\left(\boldsymbol{\varsigma}(\widetilde{V}_n)\right) = a + \frac{1 - r}{2}\log(n) + \varepsilon, \end{align} $$

where $r = k + \delta $ in which k denotes the corresponding moment order. The left side is calculated numerically from the definition in (7) with $\psi (u) = \exp (-u^2/2)$ in which we set $\theta = \theta ^*$ . The quantity in the absolute value is calculated using (5) and the explicit representations in (47) and (48). Finally, the maximization of the absolute value is conducted numerically over a grid whose size is dictated by the fast decay of $\psi (u)$ .

Both sides of (49) are evaluated for $n = 1,2,\dots , 200$ but the coefficients are estimated from the data with $n \geq 100$ because the relationship is asymptotic in theory. For the first set of parameters, we find that $\hat {r}_{\mathrm {slow}} = 1.99$ with $\hat {k}_{\mathrm {slow}} = 1$ and for the second set of parameters, $\hat {r}_{\mathrm {fast}} = 2.98$ with $\hat {k}_{\mathrm {fast}} = 2$ , which is exactly as expected. The estimated relationships together with the data are presented in Figure 4.

Figure 4

Estimated rates of convergence for a distribution that meets the third-order moment condition in (11) (in black) and for one that does not (in gray). In line with theory, the estimated rates of convergence are $\hat {r}_{\mathrm {fast}} = 2.98$ with $\hat {k}_{\mathrm {fast}} = 2$ for the data in black and $\hat {r}_{\mathrm {slow}} = 1.99$ with $\hat {k}_{\mathrm {slow}} = 1$ for the data in gray.

Finally, we compare the rate of convergence in the CLT to the rates above. It is important to note that the rate of convergence in the classical CLT depends on some additional properties of the distribution of $\epsilon $ in (44), while the rates computed above do not depend on $\epsilon $ . For example, in line with the conditions in (33), to reach the faster rate $\hat {r}_{\mathrm {fast}}$ one would need to require,

$$\begin{align*}\mathbb{E}(Y^{\ell}X^m) = \mathbb{E}(Z_y^{\ell}Z_x^m), \end{align*}$$

where $\ell ,m$ are positive integers or zero such that $\ell + m = 1,2,3$ and $(Z_y, Z_x)$ has a bivariate Gaussian distribution.Footnote ¹⁰ Clearly, these conditions imply symmetry for the marginals $\mathbb {E} Y^3 = \mathbb {E} X^3 = 0$ and X is asymmetric by construction while the symmetry of Y depends on the properties of $\epsilon $ . In this particular example, the rate behind $\hat {r}_{\mathrm {fast}}$ is not attainable for the vector $(Y, X)$ in the CLT, for example, in the Berry–Esseen setup, but even if $X \overset {\scriptscriptstyle d}{=} -X$ , then the CLT rate would depend on the properties of $\epsilon $ . As a result, the limit theorems for the conditional mean developed in this article are not a simple consequence of the CLT. Rather, they can be viewed as a separate class of results specialized for conditional means.

5.3 Beyond Lévy Processes

The focus of the article is the assumption that the data generating process follows a multivariate Lévy process. We take advantage of the property that its increments over time intervals of the equal length are i.i.d. random vectors. The proofs do not rely on the infinite divisibility of the increments and the rates of decay in Theorems 2–4 hold for sequences of i.i.d. random vectors as well.

From the standpoint of applications of discrete time processes, a more appealing setup is that of independent non-identically distributed random vectors. The framework developed in this article can handle this extension. Next, we provide a pair of assumptions similar to Assumptions 1 and 2.

Assumption 2^‡. The following moment conditions hold for all $j:$

(50) $$ \begin{align} \mathbb{E}\left((Y_{j} - X_{j}^{\mathsf{T}} \theta^*)X^{\nu}_{j}\right) &= 0, \quad \forall \nu\colon |\nu| \leq k \end{align} $$

(51) $$ \begin{align} \mathbb{E}\left(|Y_{j} - X_{j}^{\mathsf{T}}\theta^*|\|X_{j}\|_1^{r}\right) &\leq D < \infty, \end{align} $$

where $k \geq 1$ is an integer, $\nu $ denotes a multi-index, $r = k + \delta $ with $\delta \in (0, 1]$ , and $\theta ^* = {\mathbf {C}}_{XX}^{-1}{\mathbf {C}}_{YX}$ exists in which the index notation identifies the corresponding blocks in ${\mathbf {C}}$ .

Both assumptions include moment conditions holding across all members of the sequence in the same way. Such a degree of uniformity is not required in the proof but simplifies the notation and makes the connection with the i.i.d. case easier to understand. The rate of decay is provided in the following theorem.

Theorem 5. Under Assumptions 1^‡ and 2^‡ ,

(52) $$ \begin{align} \boldsymbol{\varsigma}\left(n^{-1/2}\sum_{j = 1}^nV_{j}\right) \leq C_{r}n^{-\frac{r + 1}{2}}\sum_{j = 1}^n\boldsymbol{b}_{r}(V_{j}), \end{align} $$

where $r \geq 1$ , $C_r = \sup _u\left \lVert u\right \rVert _1^r\psi (u)$ . Furthermore, the next bound holds

(53) $$ \begin{align} \sum_{j = 1}^n\boldsymbol{b}_{r}(V_{j}) \leq nA_\delta \frac{\Gamma(\delta + 1)}{\Gamma(r + 1)}D, \end{align} $$

where the notation is as in Theorem 2.

If the elements of the sequence are i.i.d., then $\boldsymbol {b}_r(V_1) = \boldsymbol {b}_r(V_j)$ for all j and it is easy to see how equations (52) and (53) turn into equations (14) and (15). Combining both inequalities in Theorem 5 indicates the rate of decay is similar to the one in Theorem 2. The main difference is the increase in the scaling constant. The similarity in the rate of decay in the two theorems follows from the conditions in equation (50) holding uniformly for all j.

Relaxing the assumption of independence is a lot more challenging and is beyond the scope of this article but it is important for applications in economics and finance (Rachev and Mittnik, Reference Rachev and Mittnik2000; Cont and Tankov, Reference Cont and Tankov2003; Stock and Watson, Reference Stock, Watson, Clements and Hendry2011). From the viewpoint of both theory and practice, one important general relaxation includes $V_1, V_2, \dots $ forming a multivariate martingale difference sequence. The CLT holds in this case subject to certain regularity conditions and rate-of-convergence results in terms of the uniform metric in discrete and continuous time can be found in Butzer, Hahn, and Roeckerath (Reference Butzer, Hahn and Roeckerath1983), Haeusler (Reference Haeusler1988), Račkauskas (Reference Račkauskas1991), and Mourrat (Reference Mourrat2013).

Another important relaxation is weak dependence which implies the correlation between the elements of the sequence decreases as the time lag increases. Rate-of-convergence results in the CLT setting for weakly dependent random variables are available, for example, in Tikhomirov (Reference Tikhomirov1991, Reference Tikhomirov1994).

Given that subject to certain regularity conditions there is a CLT result in both settings, the conditional expectations of a properly normalized sum converge to a linear function. It is a future topic of research to study the rate-of-convergence problem for conditional means in these settings and to understand how the rates compare to the corresponding CLT rates available in the academic literature.