2. Main results

Our first result establishes the finiteness of Pickands constants and (1.1) for a general RF Z under some weak restrictions. In particular, our result is satisfied for Z such that Y(t), $t\in \mathbb{R}^d$ , defined in (1.7) is a stochastically continuous stationary max-stable RF with locally bounded sample paths. Hereafter we shall suppose that all RFs are defined on a complete probability space $(\Omega, \mathcal{F}, \mathbb{P})$ . In the following, for a given stochastically continuous RF U(t), $t\in \mathcal{T}$ , we shall suppose that it is also separable and jointly measurable. In view of [Reference Doob16], a separable and jointly measurable version of a stochastically continuous RF exists.

In the following,

\begin{equation*} S_\delta (Z)= \int_{t\in \delta \mathbb{Z}^d } {Z(t)} \lambda_\delta({\mathrm{d}} t), \quad \delta\ge 0,\end{equation*}

where $\lambda_0({\mathrm{d}} t)=\lambda({\mathrm{d}} t)$ is the Lebesgue measure on $\mathbb{R}^d$ and for $\delta>0$ $\lambda_\delta({\mathrm{d}} t)= \delta ^d \lambda({\mathrm{d}} t)$ with $\lambda({\mathrm{d}} t)$ the counting measure on $\delta \mathbb{Z}^d$ .

Clearly $S_\delta(Z)$ is an RV for any $\delta>0$ . If $\delta=0$ (recall that $0\mathbb{Z}^d$ simply denotes $\mathbb{R}^d$ ), since we consider Z to be jointly measurable, then supposing that (1.5) holds and using that Z is non-negative (recall that we assume the probability space is complete), it follows that $S_0(Z)$ is an RV; see [Reference Doob16, Theorems 2.7 and 2.8] for details of the case $d=1$ .

Corollary 2.1. Suppose that Z satisfies the assumptions of Theorem 2.1 and further

\begin{equation*} \mathbb{P}\biggl\{{\sup_{t\in \mathbb{R}^d} Z(t)>0} \biggr\}=1. \end{equation*}

If $ S_\eta(Z) = \infty $ almost surely for some $\eta\ge 0$ , then $H_{Z}^\delta=0$ for all $\delta \ge 0$ . Conversely, if $H_{Z}^\delta=0$ for some $\delta \ge 0$ , then $S_\eta(Z)=\infty $ almost surely for all $\eta\ge 0$ .

In the rest of this section we consider Y(t), $t\in \mathbb{R}^d$ , a stationary max-stable RF as in Section 1 with spectral RF Z and de Haan representation (1.7). Suppose next that Z has sample paths almost surely in the space $D=D(\mathbb{R}^d, [0,\infty ))$ of generalised càdlàg functions $f\colon \mathbb{R}^d \to [0,\infty )$ equipped with the $\sigma$ -field $\mathcal{D}=\sigma(\pi_t, t\in \mathcal{T}_0)$ generated by the projection maps $\pi_t\colon \pi_t f= f(t)$ , $f\in D$ , and $\mathcal{T}_0$ a dense subset of $\mathbb{R}^d$ . See e.g. [Reference Janson21] and [Reference Bladt, Hashorva and Shevchenko4] for the definition and properties of generalised càdlàg functions. In view of [Reference Hashorva19, Theorem 6.9] (take $\alpha=1$ and $L=B^{-1}$ therein), the stationarity of Y is equivalent to the validity of

(2.1) \begin{equation} \mathbb{E}\{{ Z(h) F(Z)}\} = \mathbb{E}\{{ Z(0) F(B^h Z)}\}\end{equation}

for all $h\in \mathbb{R}^d$ and all 0-homogeneous measurable functionals $F\colon D \mapsto [0,\infty )$ (0-homogeneous means $F(cf)=F(f), \forall c>0, f\in D$ ) with $B^h Z(\cdot)= Z(\cdot-h)$ , $h\in \mathbb{R}^d$ . For discrete max-stable processes, (2.1) is stated in [Reference Planinić and Soulier30, eq. (5.2)]; see also [Reference Soulier37] for other equivalent formulations. We note in passing that (2.1) is initially derived for Z as in Example 2.1 below in [Reference Dieker and Mikosch12, Lemma 5.2].

Clearly, if Z is stationary, then Y is stationary too. However, this instance is not interesting, since as shown in Remark 2.1, (ii) $H_{Z}^0=0$ in this case. We discuss below two other examples such that max-stable RF Y is stationary.

Example 2.2. Let L(t), $t\in \mathbb{R}^d$ , be a non-negative deterministic measurable function such that $\int_{\mathbb{R}^d} {L (t)}\lambda({\mathrm{d}} t) = 1, $ with $\lambda({\mathrm{d}} t)$ the Lebesgue measure on $\mathbb{R}^d$ . It follows easily that $ Z(t) = L(t-N) / p(N) $ , $t\in \mathbb{R}^d$ , with N an $\mathbb{R}^d$ -valued RV having a positive density function $p(t)>0, t\in \mathbb{R}^d$ satisfies (1.6). Note that for this case condition (1.5) reads as

(2.2) \begin{equation} \int_{x\in \mathbb{R}^d} \sup_{t\in [0,T]^d} {L(t-x)} \lambda( {\mathrm{d}} x)< \infty \quad \text{for all } T>0.\end{equation}

As shown in [Reference Hashorva20, Theorem 1, eq. (3.5)], without any further assumption on the max-stable stationary RF Y, for $\delta=\eta> 0$ we have

(2.3) \begin{equation} H_{Z}^\delta = \mathbb{E}\biggl\{{ {Z(0)} \dfrac{\sup_{t\in \delta \mathbb{Z}^d } {Z(t)} }{S_\eta(Z) } }\biggr\} \in [0,\infty ).\end{equation}

In the next result we show that the above holds under some weak assumptions also for $\delta=0$ . It turns out that under (2.4) below we can also obtain the first formula in (1.2).

Proposition 2.1. Let Z(t), $t\in \mathbb{R}^d$ , be a non-negative RF with almost surely sample paths in D. If (1.3), (1.5), and (2.1) hold and $H_{Z}^0>0$ , then (2.3) holds for $\delta=\eta= 0$ . Moreover, (2.3) holds also for $\delta=0$ , $\eta>0$ or $\delta>0$ , $\eta=k\delta$ , $k \in \mathbb{N} $ if also

(2.4) \begin{equation} \{ S_0(Z) < \infty \} \subset \{ S_\eta(B^r Z) \in (0,\infty ) \} \quad {for\ all\ } r\in \delta \mathbb{Z}^d\end{equation}

almost surely.

Example 2.3. Let L(t), $t\in \mathbb{R}^d$ , be as in Example 2.2 and suppose further that $L(t)>0$ , $t\in \mathbb{R}^d$ , (2.2) holds and $L \in D(\mathbb{R}^d, [0,\infty ))$ . Then the claim of Theorem 2.1 follows. Moreover, Proposition 2.1 implies for all $\eta >0$

(2.5) \begin{equation} H_{Z}^0=\int_{\mathbb{R}^d} L (s)\dfrac{ \sup_{t\in \mathbb{R}^d} {L(t+s)} } { \eta^d \sum_{t\in \eta \mathbb{Z}^d} {L(t+s)} } \lambda({\mathrm{d}} s)= \sup_{t\in \mathbb{R}^d} {L (t) } ,\end{equation}

where the last equality follows by (2.3) with $\delta=0$ (recall that $\int_{\mathbb{R}^d} L(t) \lambda({\mathrm{d}} t)=1$ ). It is mentioned in [Reference Dieker and Yakir13] that for $d=1$ and L the standard Gaussian density on $\mathbb{R}$ , the above identity could be verified numerically. As shown by Dmitry Zaporozhets (personal communications) the last two equalities in (2.5) can be derived by utilising the translation invariance of both the Lebesgue measure and the counting measure, respectively, and applying the Fubini–Tonelli theorem. Specifically, setting $\eta=1$ for simplicity,

\begin{align*}\int_{\mathbb{R}^d}\dfrac{ L (s) } { \sum_{t\in \mathbb{Z}^d} {L(t+s)} } \lambda({\mathrm{d}} s) &=\sum_{ r \in \mathbb{Z}^d} \int_{ r+ [0,1)^d }\dfrac{ L(s) } { \sum_{t\in \mathbb{Z}^d} {L(t+s)} } \lambda({\mathrm{d}} s)\\ &= \sum_{ r \in \mathbb{Z}^d} \int_{ [0,1)^d }\dfrac{ L(r+s) } { \sum_{t\in \mathbb{Z}^d} {L(r+t+s)} } \lambda({\mathrm{d}} s)\\ &= \int_{ [0,1)^d }\sum_{ r \in \mathbb{Z}^d}\dfrac{ L(r+s) } { \sum_{t\in \mathbb{Z}^d} {L(t+s)} } \lambda({\mathrm{d}} s)=1.\end{align*}

Our initial proof of Proposition 2.1 was asymptotic in nature. A modification of Zaporozhets’ arguments and (2.1) led to the current proof of Proposition 2.1.

Note that if we take for instance $L(t)=\textbf{1}{\{{t\in [0,1]}\}}$ , $d=1$ , with $\textbf{1}{\{{\cdot}\}}$ the indicator function, then $H_Z^0$ cannot be given by (2.3) with $\eta>2$ , since the first equality in (2.5) gives $H_Z^0=\infty $ , which is a contradiction to the fact that $H_Z^ 0< \infty $ . For this choice of $\eta$ we have that condition (2.4) is not satisfied.

3. Two extensions

Motivated by [Reference Samorodnitsky36] and [Reference Samorodnitsky35], we first consider an alternative definition of Pickands constants $H_{\lvert {f} \rvert,m}^\delta $ defined with respect to a collection of functions $f_s\colon E \to \mathbb{R} $ , with $(E, \mathcal{E})$ a measurable space equipped with some $\sigma$ -finite measure m. In the second part of this section we discuss the constant $\mathcal{H}_{Z}^\delta$ defined in (1.10), also proving its continuity at $\delta=0$ .

3.1. Behaviour of $\boldsymbol{H}_{\lvert \boldsymbol{f} \rvert\boldsymbol{,m}}^{\boldsymbol\delta} $ at $\delta=0$

Let m be a $\sigma$ -finite measure on some measurable space $(E, \mathcal{E})$ and $f=f_s(z)$ , $s\in \mathbb{R}^d$ , $z\in E$ real functions, where $f_s \in \mathcal{L}^1 (m)$ ( $\mathcal{L}^\alpha(m)$ is the set of all functions g such that $\int_{E}|g(x)|^\alpha m( {\mathrm{d}} x)<\infty$ and $\alpha>0$ ). We now define for any $\delta =1/n$ , $n \in \mathbb{N} $ , or $\delta=0$

\begin{equation*} H_{\lvert {f} \rvert,m}^\delta = \lim_{{T} \to \infty} \dfrac{1}{T^d}\int_{E } \sup_{t\in [0,T]^d \cap \delta \mathbb{Z}^d} \lvert {f_t (z)} \rvert m( {\mathrm{d}} z),\end{equation*}

where $0\mathbb{Z}^d$ equals $\mathbb{Q}^d_*$ , with $\mathbb{Q}_*$ the set of dyadic rational numbers $\{{k}/{2^n}\colon k\in \mathbb{Z}, n\in \mathbb{N} \}$ . Clearly the above limit is in general not defined. In order to include the case $\delta=0$ , we shall assume further that

(3.1) \begin{equation} \int_E \sup_{t\in K \cap \mathbb{Q}_*^d} \lvert {f_t(z)} \rvert m( {\mathrm{d}} z) < \infty \end{equation}

and, similarly to (1.6), we have

(3.2) \begin{equation} \int_E \sup_{t\in K \cap \mathbb{Q}_*^d } \lvert {f_{t+c}(z)} \rvert m( {\mathrm{d}} z) = \int_E \sup_{t\in K \cap \mathbb{Q}_*^d } \lvert {f_{t}(z)} \rvert m( {\mathrm{d}} z) \quad\text{for all } c\in \mathbb{Q}_*^d\end{equation}

for all compact sets $K \subset \mathbb{R}^d$ .

If L is as in Example 2.2, then taking $f_t(z)= L(z-t)$ , $t, z\in \mathbb{R}^d$ , $E=\mathbb{R}^d$ equipped with the Borel $\sigma$ -field and $m( {\mathrm{d}} z)$ the Lebesgue measure on $\mathbb{R}^d$ , we have that (3.1) holds and further (3.2) is satisfied since $m( {\mathrm{d}} z)$ is shift-invariant.

Example 3.1. Let Y(t), $t\in \mathbb{R}^d$ , be a symmetric $\alpha$ -stable stationary RF with locally bounded sample paths, $\alpha \in (0,2)$ , and representation

\begin{equation*} Y(t) = \int_E f_t(z) M( {\mathrm{d}} z) , \quad t\in \mathbb{R}^d , \end{equation*}

where M is a symmetric $\alpha$ -stable random measure on E with control measure m and $f_t\in \mathcal{L}^\alpha(m)$ , $t\in \mathbb{R}^d$ ; see [Reference Samorodnitsky36]. It follows that both (3.1) and (3.2) hold; see [Reference Samorodnitsky36] for the case $d=1$ and [Reference Roy33] for the case $d>1$ .

With the same arguments as in the proof of Theorem 2.1, we have that under (3.1) and (3.2), constant $H_{\lvert {f} \rvert,m}^\delta $ is finite, non-negative, and further

(3.3) \begin{equation} \lim_{{n} \to \infty} H_{\lvert {f} \rvert, m}^{2^{-n}} =H_{\lvert {f} \rvert, m}^{0} .\end{equation}

Remark 3.1. For Y as in Example 3.1, explicit formulas for $H_{\lvert {f} \rvert, m}^{\delta} $ are derived in [Reference Samorodnitsky36], [Reference Samorodnitsky35], and [Reference Roy and Samorodnitsky34]. Utilising the relation between $\alpha$ -stable and max-stable processes (see [Reference Kabluchko22]), the aforementioned formulas imply (2.3) when $\delta=\eta\ge 0$ .

3.2. Continuity of $\mathcal{H}_Z^\delta$ at $\delta=0$

Let $W_z(t)$ , $t\in \mathbb{R}^d$ , $z\in [0,1] $ , be a centered Gaussian RF with stationary increments, almost surely continuous sample paths, and variance function $\sigma^2_z$ , $z\in [0,1]$ , such that $\sigma^2_z(0)=0$ for all $z\in [0,1].$ We formulate below the assumptions on the variance functions $\sigma_z^2$ imposed in Theorem 3.1 below. Specifically, we shall assume that

(3.4) \begin{equation} \lim_{w\to z}\sigma_w(t)=\sigma_z(t) \quad \text{for all } z \in [0,1], \ t\in \mathbb{R}^d\end{equation}

and further, for some $C_{0},C_{\infty}$ positive and $\nu_{0},\nu_{\infty}\in (0,2] $ ,

(3.5) \begin{equation} \limsup_{\lVert {t} \rVert\to 0} \dfrac{ \sigma_z^2(t)}{\lVert {t} \rVert^{\nu_{0}}} \le C_{0}\quad \text{and}\quad\limsup_{\lVert {t} \rVert\to \infty} \dfrac{ \sigma_z^2(t)}{\lVert {t} \rVert^{\nu_{\infty}}} \le C_{\infty}\end{equation}

hold for all $z\in [0,1]$ , where $\lVert {\cdot} \rVert$ denotes the Euclidean norm on $\mathbb{R}^d$ .

Theorem 3.1. If (3.4) and (3.5) are satisfied, then for any $\delta\geq 0$ and $Z_z(t)= \exp( W_z(t)- \sigma_z^2(t)/2)$ ,

(3.6) \begin{equation}\mathcal{H}_{{{Z}}}^\delta = \lim_{{T} \to \infty} T^{-d} \int_0^1 \mathbb{E}\biggl\{{\sup_{ t\in [0,T]^d \cap \delta \mathbb{Z}^d }Z_z (t) }\biggr\} \,{\mathrm{d}} z=\int_0^1 H_{Z_z}^\delta \,{\mathrm{d}} z \in [0,\infty) ,\end{equation}

and furthermore $\lim_{\delta \downarrow 0} \mathcal{H}_{Z}^\delta= \mathcal{H}_{Z}^0$ .

Remark 3.2.

1. (i) A sufficient condition for $H_{Z_z}^\delta, z\in [0,1]$ to be positive is

   (3.7) \begin{equation}\lim_{{\lVert {t} \rVert} \to \infty} \dfrac{ \sigma_z^2(t)}{ \ln \lVert {t} \rVert} > 8 d\end{equation}
   (see [Reference Dębicki and Hashorva9]). If the above holds for $z\in [0,1]$ on a set with non-zero Lebesgue measure, then under the assumptions of Theorem 3.1 we have that $\mathcal{H}_{Z}^\delta>0$ for any $\delta \ge 0$ .

2. (ii) In order to define RFs $W_z,z\in [0,1]$ with stationary increments, we need to determine variance function $\sigma^2_z(t)= \textrm{Var}( W_z(t+s)- W_z(t))$ , $s,t\in \mathbb{R}^d$ , which is a negatively (or conditionally negatively) definite function. One example is to start with a negatively definite variance function $\sigma^2(t)$ , $t\in \mathbb{R}^d$ , and then define $\sigma_z(t)= q(z) \sigma(t)$ for some continuous function q not identical to zero. Such $\sigma_z$ clearly satisfies (3.4) and if $\sigma^2(t)\le C \lVert {t} \rVert^\nu$ , $t\in \mathbb{R}^d$ , then (3.5) also holds. Another example is to consider $\sigma_z$ as in [Reference Chan and Lai5], namely

   (3.8) \begin{equation}\sigma^2_z(t) = \lVert {t} \rVert^\lambda r_z(t/\lVert {t} \rVert), \quad \lambda \in (0,2], \ t\in \mathbb{R}^d,\end{equation}
   where $r_z$ is a non-negative function defined on the unit sphere on $\mathbb{R}^d$ determined by the norm $\lVert {\cdot} \rVert$ .

4. Proofs

Proof of Theorem 2.1. In view of (1.5) and (1.6),

(4.1) \begin{equation}\mathbb{E}\biggl\{{ \sup_{t\in K \cap \delta \mathbb{Z}^d} Z(t+c)}\biggr\}= \mathbb{E}\biggl\{{ \sup_{t\in K \cap \delta \mathbb{Z}^d} Z(t)}\biggr\}< \infty\end{equation}

holds for all $c\in \mathbb{R}^d$ , $\delta \ge 0$ , and all compact sets $K\subset \mathbb{R}^d$ . Next, assume for notational simplicity that $d=1$ . In light of (4.1), $\mathbb{E}\{{ \sup_{t\in [0,T] \cap \delta \mathbb{Z} } {Z(t)}}\}$ , $T>0$ , is a subadditive function of T for any $\delta\ge 0$ , implying that $H_Z^\delta$ is a finite non-negative constant (recall that Z is non-negative) in view of Fekete’s lemma. Further, to show the second part of the theorem it is enough to take $T= a_\delta n$ , where $a_\delta=\lfloor 1/\delta \rfloor \delta $ with $\lfloor x \rfloor$ the integer part of $x>0$ and $n \in \mathbb{N} $ $\delta \in (0,1/2)$ . Utilising (4.1) to derive the last equality in the next calculations, we have

(4.2) \begin{align} & \mathbb{E}\biggl\{{ \sup_{t\in [0,T]} {Z(t)} }\biggr\}-\mathbb{E}\biggl\{{ \max_{t\in [0,T] \cap \delta \mathbb{Z}} {Z(t)} }\biggr\}\nonumber\\&\quad = \mathbb{E}\biggl\{{ \max_{ 1 \le i \le n} \sup_{t\in [a_\delta(i-1),a_\delta i]} {Z(t)} - \max_{ 1 \le i \le n} \max_{t\in [a_\delta(i-1),a_\delta i] \cap \delta \mathbb{Z}} {Z(t)} }\biggr\}\nonumber\\&\quad \leq \mathbb{E}\biggl\{{ \max_{ 1 \le i \le n} \biggl(\sup_{t\in [a_\delta(i-1),a_\delta i]} {Z(t)} - \max_{t\in [a_\delta(i-1),a_\delta i] \cap \delta \mathbb{Z}} {Z(t)} \biggr) }\biggr\}\nonumber\\&\quad \le \sum_{i=1}^n \mathbb{E}\biggl\{{ \sup_{t\in [a_\delta(i-1),a_\delta i]} {Z(t)} - \max_{t\in [a_\delta(i-1),a_\delta i] \cap \delta \mathbb{Z}} {Z(t)} }\biggr\}\nonumber\\&\quad = n \mathbb{E}\biggl\{{ \sup_{t\in [0,a_\delta]} {Z(t)} - \max_{t\in [0,a_\delta] \cap \delta \mathbb{Z}} {Z(t)} }\biggr\}.\end{align}

The assumption that Z is stochastically continuous implies that any dense subset of $\mathbb{R}^d$ is a separant for the separable RF Z (see e.g. [Reference Potthoff31, Theorem 2.9]) and therefore, since $\lim_{\delta \downarrow 0} a_\delta=1$ , we have the following convergence in probability:

\begin{equation*}\max_{t\in [0,a_\delta] \cap \delta \mathbb{Z}} {Z(t)} \to \sup_{t\in [0,1]} {Z(t)}\end{equation*}

as $\delta \downarrow 0$ . Hence, by (1.5) and the dominated convergence theorem,

(4.3) \begin{equation} \lim_{\delta \downarrow 0 } \mathbb{E}\biggl\{{ \max_{t\in [0,a_\delta ] \cap \delta \mathbb{Z}} {Z(t)} }\biggr\}=\mathbb{E}\biggl\{{ \sup_{t\in [0,1]} {Z(t)}}\biggr\}< \infty .\end{equation}

Consequently (4.2) implies

\begin{align*} \dfrac{1}{T} \mathbb{E}\biggl\{{ \max_{t\in [0,T] \cap \delta \mathbb{Z}} {Z(t)}}\biggr\} & \leq \dfrac{1}{T} \mathbb{E}\biggl\{{ \sup_{t\in [0,T]} {Z(t)}}\biggr\}\\ &\leq a_\delta^{-1} \mathbb{E}\biggl\{{ \sup_{t\in [0,a_\delta]} {Z(t)} - \max_{t\in [0,a_\delta] \cap \delta \mathbb{Z}} {Z(t)} }\biggr\}+ \dfrac{1}{T}\mathbb{E}\biggl\{{ \max_{t\in [0,T] \cap \delta \mathbb{Z}} {Z(t)}}\biggr\}.\end{align*}

Since $H^\delta_Z$ exists and is finite for all $\delta \ge 0$ , we have

\begin{equation*} \lim_{{T} \to \infty} \dfrac{1}{T} \mathbb{E}\biggl\{{ \sup_{t\in [0,T]} {Z(t)}}\biggr\} = \lim_{{n} \to \infty} \dfrac{1}{ n a_\delta } \mathbb{E}\biggl\{{ \sup_{t\in [0,n a_\delta]} {Z(t)}}\biggr\} = H^0_Z\end{equation*}

and

\begin{equation*} \lim_{{T} \to \infty} \dfrac{1}{T} \mathbb{E}\biggl\{{ \sup_{t\in [0,T] \cap \delta \mathbb{Z} } {Z(t)}}\biggr\} =\lim_{{n} \to \infty} \dfrac{1}{ n a_\delta }\mathbb{E}\biggl\{{ \sup_{t\in [0,n a_\delta] \cap \delta Z} {Z(t)}}\biggr\} = H^\delta _Z \le H^0_Z<\infty .\end{equation*}

Thus, letting n to infinity, we obtain from the above inequalities

(4.4) \begin{equation}0 \leq H_{Z}^{0 }-H_{Z}^{\delta } \leq a_\delta^{-1} \mathbb{E}\biggl\{{ \sup_{t\in [0,a_\delta]} {Z(t)} - \max_{t\in [0,a_\delta] \cap \delta \mathbb{Z}} {Z(t)} }\biggr\},\end{equation}

which together with (4.3) yields $\lim_{\delta \downarrow 0} H_{Z}^{\delta }= H_{Z}^0$ .

Proof of Corollary 2.1. First, let us notice that $H_{Z^{}}^\delta =0$ for some $\delta\geq 0$ if and only if $S_\delta(Z)=\infty $ almost surely, which is a direct implication of [Reference Dombry and Kabluchko14], [Reference Kabluchko22], and [Reference Roy and Samorodnitsky34, Reference Samorodnitsky35, Reference Samorodnitsky36] ; this has already been discussed in [Reference Ehlert and Schlather17], [Reference Dębicki and Hashorva9], and [Reference Soulier37] for the case $d=1$ , and [Reference Hashorva20] for the d-dimensional discrete setup. Note in passing that in [Reference Dombry and Kabluchko14] Z is such that $\mathbb{P} \{ {\sup_{t\in \mathbb{R}^d} Z(t)> 0} \} =1$ . So if for some $\eta>0$ we have $H_{Z}^\eta=0$ , then $S_{\eta/k}( Z) = \infty $ almost surely for any $k\in N$ and $H_{Z }^{\eta/k}=0$ . Consequently, Theorem 2.1 implies that $\lim_{{k} \to \infty}H_{Z }^{\eta/k}=H_{Z }^0=0$ and hence $0\le H_{Z}^\eta\le H_{Z }^{0}= 0$ for all $\eta\ge 0$ . Conversely, if for some $\eta\ge 0$ we have $S_{\eta}( Z) = \infty $ , then from the above equivalence $H_{Z}^\eta=0$ for all $\eta\ge 0$ , hence the proof is complete.

Proof of Proposition 2.1. The claim for $\delta=\eta=0$ follows as in [Reference Dębicki and Hashorva9], where $d=1$ is considered and its proof is therefore omitted.

Now let $D=D(\mathbb{R}^d, [0,\infty ))$ be the space of generalised càdlàg functions $f\colon \mathbb{R}^d \mapsto [0,\infty )$ , which can be equipped with a metric that turns it into a Polish space. The corresponding Borel $\sigma$ -field in D denoted by $\mathcal{D}$ agrees with the $\sigma$ -field $\sigma(\pi_t, t\in \mathcal{T}_0)$ for any $\mathcal{T}_0$ a dense subset of $\mathbb{R}^d$ ; see e.g. [Reference Bladt, Hashorva and Shevchenko4, Theorem 7.1]. Let $\Theta(t)$ , $t\in \mathbb{R}^d$ , denote an RF with almost surely sample paths in D defined by

\begin{equation*}\mathbb{P} \{ { \Theta \in A} \} = \mathbb{E}\{{ Z(0) \textbf{1} ( Z/Z(0)\in A)}\}/\mathbb{E}\{{ Z(0)}\} \quad \text{for all } A \in \mathcal{D},\end{equation*}

where we interpret $0:0$ as 0 and set $\textbf{1} ( x\in A)$ equal to 1 or 0 if $x\in A$ or $x\not \in A$ , respectively. Note that $\mathbb{E}\{{ Z(0)}\}$ equals 1, but we leave it since the same formula is applied to $Z_*$ below. Note further that since D is Polish, by [Reference Varadarajan38, lemma on page 1276] we can realise both Z and $\Theta$ in the same complete non-atomic probability space that we assume for notational simplicity below. By (2.1), for any measurable functional $F\colon D \mapsto [0,\infty ]$ that is 0-homogeneous,

(4.5) \begin{equation} \mathbb{E}\{{ \Theta (h) F(\Theta)}\} =\mathbb{E}\{{ \textbf{1} ( Z(0)\not=0) Z (h) F(Z)}\}= \mathbb{E}\{{ \textbf{1} (\Theta(-h) \not=0 ) F(B^h \Theta) } \}.\end{equation}

In view of Corollary 2.1, the assumption $H^0_{Z}>0$ implies $\mathbb{P} \{ {S_\eta(Z)< \infty } \} =p_\eta >0$ for all $\eta\ge 0$ . If $p_0=1$ , then $\mathbb{P} \{ {S_0(\Theta) < \infty } \} =1$ also follows, and thus

(4.6) \begin{equation} \{ S_\eta(\Theta) < \infty \} \subset \{ S_0(\Theta)< \infty \} \end{equation}

almost surely.

Assume next that $p_0\in (0,1)$ . The RF $Z_*(t)=Z(t) \mid S_0(Z) = \infty , \,t\in \mathbb{R}^d $ has almost surely sample paths in D and satisfies (2.1). Let $\Theta_*$ denote the corresponding RF of $Z_*$ defined by the change of measure as above. We have that $\Theta_*$ has the same law as $\Theta \mid S_0(\Theta)= \infty $ . Since $ S_0( Z_*) =\infty $ with probability 1, applying Corollary 2.1, we obtain $ S_\eta(Z_*)=\infty $ almost surely and thus $S_\eta (\Theta_*)=\infty $ almost surely, that is,

\begin{equation*}\mathbb{P} \{ { S_\eta(\Theta)=\infty , S_0(\Theta)= \infty } \} /\mathbb{P} \{ {S_0(\Theta)= \infty } \} =1\end{equation*}

(implying $ \{ S_0(\Theta) = \infty \} \subset \{ S_\eta(\Theta) = \infty \}$ almost surely), and hence (4.6) holds. If $\mathbb{P} \{ {S_0(Z)< \infty } \} =0$ , then by Corollary 2.1 $\mathbb{P} \{ {S_\eta(Z)< \infty } \} =0$ for all $\eta>0$ , which in turn implies $\mathbb{P} \{ { S_\eta(\Theta) < \infty } \} =0$ . The assumption $p_0=0$ also implies $\mathbb{P} \{ { S_0(\Theta) < \infty }\} =0$ , hence again (4.6) holds.

Next suppose that $\mathbb{P} \{ {S_0(\Theta)< \infty } \} >0$ . Since then $\mathbb{P} \{ {S_0(Z)< \infty } \} >0$ , we can define $Z_*(t)=Z(t) \mid S_0(Z) < \infty $ , $ t\in \mathbb{R}^d $ , as above. The corresponding $\Theta_*$ has the same law as $\Theta \mid S_0(\Theta)< \infty $ . Since $\mathbb{P} \{ {S_0(Z_*)< \infty } \} =\mathbb{P} \{ {S_0(\Theta_*)< \infty } \} =1$ , by [Reference Soulier37, Theorem 2.8],

\begin{equation*}\mathbb{P}\biggl\{{\lim_{\lVert {t} \rVert\to \infty , t\in \eta \mathbb{Z}^d} \Theta_*(t) =0}\biggr\}=1,\end{equation*}

with $\lVert {\cdot} \rVert$ some norm on $\mathbb{R}^d$ . The latter is equivalent to $\mathbb{P} \{ {S_\eta(\Theta_*)< \infty } \} =1$ ; see [Reference Hashorva20, Condition A2, A4]. As in the proof of [Reference Hashorva20, Lemma A.2], $\mathbb{P} \{ { S_0(\Theta) < \infty } \} =0 \iff p_0=0$ and thus the reverse inclusion to (4.6) holds, implying

(4.7) \begin{equation}\{ S_\eta(\Theta) < \infty \} =\{ S_0(\Theta)< \infty \}\quad \text{for all } \eta>0\end{equation}

almost surely. Since further $\mathbb{P} \{ {S_\eta(\Theta)>0} \} =1$ for all $\eta\ge 0$ , which follows from $\mathbb{P} \{ {\Theta(0)=1} \} =1$ and the fact that $\Theta$ has paths in D almost surely, we have almost surely for all $\delta,\eta \in [0,\infty )$

(4.8) \begin{equation} \dfrac{ 1 }{S_\eta(\Theta)} =\dfrac{ 1 }{S_\eta(\Theta)} \dfrac{ S_\delta(\Theta)} {S_\delta(\Theta)}\Theta(0).\end{equation}

Now set $S_\eta(f)= \int_{ \eta \mathbb{Z}^d} f (t)\lambda_\eta({\mathrm{d}} t)$ , $\eta\ge 0$ , and

\begin{equation*}M_\eta(f)= \sup_{t\in \eta \mathbb{Z}^d} f (t), \quad M_0(f)= \sup_{t\in \mathcal{T}_0} f(t),\quad f\in D,\ \eta >0. \end{equation*}

Both maps $S_\eta(\cdot)$ and $M_\eta(\cdot)$ , $\eta \ge 0$ , are measurable and by the separability of Z we have that $M_0(Z)$ has the same law as $\sup_{t\in \mathbb{R}^d} Z(t)$ . Hence, also using the definition of $\Theta$ and (4.8), by the Fubini–Tonelli theorem, for all $\delta\not=\eta$ , $\delta \ge 0$ , $\eta>0$ we have

(4.9) \begin{align} & \mathbb{E}\biggl\{{ Z (0) \dfrac{ \sup_{t\in \delta \mathbb{Z}^d} Z (t)}{S_{\eta }(Z)}}\biggr\}\notag\\[4pt]&\quad = \mathbb{E}\biggl\{{ \dfrac{ \sup_{t\in \delta \mathbb{Z}^d} \Theta (t)}{S_{\eta }(\Theta)}}\biggr\} \notag\\[4pt]&\quad = \mathbb{E}\biggl\{{ \dfrac{ \sup_{t\in \delta \mathbb{Z}^d} \Theta (t)}{S_{\eta }(\Theta)} \dfrac{ S_\delta(\Theta)}{ S_\delta(\Theta)} \Theta(0)}\biggr\} \notag\\[4pt]&\quad = \int_{\delta \mathbb{Z}^d}\mathbb{E}\biggl\{{ \dfrac{ M_\delta ( \Theta)}{ S_\delta(\Theta)}\dfrac{ \Theta (0)}{ S_{\eta }(\Theta) } \Theta (s) }\biggr\} \lambda_\delta ({\mathrm{d}} s) \notag\\[4pt]&\quad = \sum_{i \in \eta \mathbb{Z}^d} \int_{r\in [0,\eta)^d \cap \delta \mathbb{Z}^d }\mathbb{E}\biggl\{{ \dfrac{ M_ \delta (\Theta) }{ S_\delta(\Theta) }\dfrac{ \Theta (0)}{ S_{\eta }(\Theta)} \Theta (i+r) }\biggr\} \lambda_\delta ({\mathrm{d}} r), \end{align}

where in the last equality we also used the translation invariance of $\lambda_\delta$ . Now, by (4.5), for all $i,r \in \mathbb{R}^d$ ,

(4.10) \begin{equation}\mathbb{E}\biggl\{{ \Theta (i+r) \dfrac{M_ \delta (\Theta) }{ S_\delta(\Theta) }\dfrac{ \Theta (0)}{ S_{\eta }(\Theta)}}\biggr\} = \mathbb{E}\biggl\{{ \dfrac{\ M_ \delta (B^{r+i}\Theta)}{ S_\delta(B^{r+i}\Theta)} \dfrac{\Theta (-i-r) }{S_{\eta }(B^{r+i}\Theta)} }\biggr\}.\end{equation}

Note in passing that $\Theta(-i-r) \textbf{1} (\Theta(-i-r) \not=0 )=\Theta(-i-r) $ almost surely.

If $\delta=0$ , $\eta>0$ or $\delta>0$ and $\eta= k\delta$ , $k\in \mathbb{N} $ , then almost surely, for all $i \in \eta \mathbb{Z}^d, r\in \delta \mathbb{Z}^d$ ,

(4.11) \begin{equation} M_ \delta (B^{r+i}\Theta)= M_ \delta (\Theta), \quad S_\delta(B^{r+i}\Theta)= S_\delta(\Theta), \quad S_{\eta }(B^{r+i}\Theta) =S_{\eta }(B^{r}\Theta),\end{equation}

and from (2.4)

\begin{align*} S_\delta (\Theta) < \infty &= \mathbb{P} \{ {S_0 (\Theta) < \infty } \} \\[3pt]&= \mathbb{E}\{{ Z(0) \textbf{1}{\{{ S_0(Z)< \infty }\}}}\}\\[3pt]&= \mathbb{E}\{{ Z(0) \textbf{1}{\{{ S_0(Z)< \infty , S_\eta(B^r Z) \in (0,\infty ) }\}}}\}\\[3pt]&= \mathbb{E}\{{ Z(0) \textbf{1}{\{{ S_{\delta}(Z)< \infty , S_\eta(B^r Z) \in (0,\infty ) }\}}}\}\\[3pt]&= \mathbb{E}\{{ Z(0) \textbf{1}{\{{ S_{\delta}(Z/Z(0))< \infty , S_\eta(B^r Z/Z(0)) \in (0,\infty ) }\}}}\}\\[3pt]&= \mathbb{P} \{ { S_\delta(\Theta) < \infty , S_\eta(B^r \Theta) \in (0,\infty )} \} \quad \text{for all } r\in \delta \mathbb{Z}^d,\end{align*}

where the third-to-last line follows from (4.7). Consequently, almost surely

(4.12) \begin{equation} \dfrac {S_{\eta }(B^r\Theta) }{S_{\eta }(B^r\Theta)} \textbf{1}{\{{S_\delta(\Theta) <\infty }\}} = \textbf{1}{\{{S_\delta(\Theta) <\infty }\}} .\end{equation}

Hence, for these choices of $\delta $ and $\eta$ , (4.9)–(4.12) and the Fubini–Tonelli theorem yield

\begin{align*} \mathbb{E}\biggl\{{ Z(0) \dfrac{ \sup_{t\in \delta \mathbb{Z}^d} Z(t)}{S_{\eta }(Z)}}\biggr\}&= \int_{r\in [0,\eta)^d \cap \delta \mathbb{Z}^d }\mathbb{E}\biggl\{{ \dfrac{\ M_ \delta (\Theta)}{ S_\delta(\Theta)} \sum_{i \in \eta \mathbb{Z}^d} \dfrac{\Theta (-r-i) }{S_{\eta }(B^r\Theta)} }\biggr\} \lambda_\delta ({\mathrm{d}} r) \\[4pt]&= \eta^{-d} \int_{r\in [0,\eta)^d \cap \delta \mathbb{Z}^d }\mathbb{E}\biggl\{{ \dfrac{\ M_ \delta (\Theta)}{ S_\delta(\Theta)} \dfrac {S_{\eta }(B^r\Theta) }{S_{\eta }(B^r\Theta)} \textbf{1}{\{{S_\delta(\Theta) <\infty }\}} }\biggr\} \lambda_\delta ({\mathrm{d}} r) \\[4pt]&= \eta^{-d} \int_{r\in [0,\eta)^d \cap \delta \mathbb{Z}^d }\mathbb{E}\biggl\{{ \dfrac{ M_ \delta (\Theta)}{ S_\delta(\Theta)} \textbf{1}{\{{S_\delta(\Theta) <\infty }\}} } \biggr\} \lambda_\delta ({\mathrm{d}} r) \\[4pt]&= \mathbb{E}\biggl\{{ \dfrac{ M_ \delta (\Theta)}{ S_\delta(\Theta)} }\biggr\}\\[4pt]&= \mathbb{E}\biggl\{{ Z(0)\dfrac{\sup_{t\in \delta \mathbb{Z}^d} Z(t)}{ S_\delta(Z)} }\biggr\}, \end{align*}

and hence the claim follows from (2.3).

Note that if $\mathbb{P} \{ {Z(0)>0} \} =1$ , by (2.1) for all $ t\in \mathbb{R}^d$ ,

\begin{equation*} \mathbb{E}\{{ Z(t)}\}=\mathbb{E}\{{ Z(t) \textbf{1}{\{{Z(0) >0}\}}}\}=\mathbb{E}\{{ Z(0) \textbf{1}{\{{Z(-t) >0}\}}}\} =\mathbb{E}\{{Z(0)}\}, \end{equation*}

and hence $\mathbb{P} \{ {Z(t)>0} \} =\mathbb{P} \{ {\Theta(t)>0} \} =1 $ since $\mathbb{E}\{{Z(t)}\}=\mathbb{E}\{{Z(0)}\}=1$ . Consequently, applying (4.5) and utilising (4.7) for all $r\in \mathbb{R}^d$ , $\eta>0$ ,

\begin{align*}& \mathbb{E}\{{ Z(-r) \textbf{1}{\{{ S_0(Z) < \infty , 0< S_\eta(B^r Z) < \infty }\}}}\}\\ &\quad =\mathbb{E}\{{ \Theta(-r)\textbf{1}{\{{ S_0(\Theta) < \infty , 0< S_\eta(B^r\Theta) < \infty }\}}}\}\\&\quad = \mathbb{E}\{{ \textbf{1}{\{{ S_0(\Theta) < \infty , 0< S_\eta(\Theta) < \infty }\}}}\}\\ &\quad = \mathbb{E}\{{\textbf{1}{\{{ S_0(\Theta) < \infty }\}}}\}\\ &\quad= \mathbb{E}\{{ Z(0) \textbf{1}{\{{ S_0(Z) < \infty }\}}}\}\\ &\quad = \mathbb{E}\{{ Z(-r) \textbf{1}{\{{ S_0(Z) < \infty }\}}}\}\\ &\quad \le 1,\end{align*}

and thus (2.4) follows. The reason that the indicator function did not appear when we applied (4.5) in the above calculations is that $\Theta(t)>0$ almost surely for all $t\in \mathbb{R}^d$ .

Proof of Theorem 3.1. Let $z\in [0,1]$ be fixed. For simplicity we consider the case $d=1$ and set $X_z(t)=W_z(t)-\sigma^2_z(t)/2$ . We now show that for any positive T the function

\begin{equation*} A_T(z)=\mathbb{E}\biggl\{{ \sup_{t\in [0,T] \cap \delta \mathbb{Z}} \,{\mathrm{e}}^{ X_{z}(t)}}\biggr\}\end{equation*}

is continuous in z and thus integrable for any $\delta\ge 0$ . First note that, by (1.5),

\begin{equation*} \mathbb{E}\biggl\{{ \sup_{t\in [0,T] \cap \delta \mathbb{Z}} \,{\mathrm{e}}^{ X_{z}(t)}}\biggr\} =1+ \int_{0}^\infty \,{\mathrm{e}}^s \mathbb{P}\biggl\{{ \sup_{t\in [0,T] \cap \delta \mathbb{Z}} X_{z}(t)> s} \biggr\} \,{\mathrm{d}} s< \infty .\end{equation*}

Since $\sigma_z$ determines the covariance function of $X_{z }$ , then (3.4) implies that FIDIs of $X_{z+h }$ converge weakly to those of $X_z$ as $h\to 0$ . Moreover, for some $\varepsilon >0$ and all $ \lvert {h} \rvert< \varepsilon $ , by (3.5),

\begin{equation*} \textrm{Var} ( X_{z+h} (s)- X_{z+h}(t))= \sigma^2_{z+h}(t-s) \le C \lVert {t-s} \rVert^{\nu_{0}} \end{equation*}

for some $C>0$ and all $t,s \in [0,T]$ . Consequently, $X_{z+h}(t)$ , $t\in [0,T]$ , is tight (with respect to h) and converges weakly in the space of real-valued continuous functions on [0,T] equipped with the uniform topology; see [Reference Piterbarg29, Proposition 9.7]. Hence, by the continuous mapping theorem for almost all $s \in \mathbb{R}$ ,

\begin{equation*} \lim_{ h \to 0} \mathbb{P}\biggl\{{ \sup_{t\in [0,T] \cap \delta \mathbb{Z}} X_{z+h}(t)> s} \biggr\} =\mathbb{P}\biggl\{{ \sup_{t\in [0,T] \cap \delta \mathbb{Z}} X_{z}(t)> s} \biggr\}.\end{equation*}

Consequently, for all $T>0$ ,

\begin{equation*} \lim_{ h \to 0}\mathbb{E}\biggl\{{ \sup_{t\in [0,T] \cap \delta \mathbb{Z}} \,{\mathrm{e}}^{ X_{z+h}(t)}}\biggr\} =\mathbb{E}\biggl\{{ \sup_{t\in [0,T] \cap \delta \mathbb{Z}} \,{\mathrm{e}}^{ X_{z}(t)}}\biggr\} =A_T(z)< \infty ,\end{equation*}

implying the claim that $A_T(z)$ , $z\in [0,1]$ , is continuous in z. The derivation above rests on the application of the dominated convergence theorem that can be justified by the Borell–TIS inequality. Indeed, for all $z\in [0,1]$ ,

\begin{equation*}\sup_{t\in [0,T] \cap \delta \mathbb{Z}} \,{\mathrm{e}}^{ X_{z}(t)}\leq \sup_{t\in[0,T], v\in[0,1]}\,{\mathrm{e}}^{W_v(t)}\end{equation*}

and

\begin{equation*}\mathbb{E}\biggl\{{ \sup_{t\in [0,T], v\in[0,1]} \,{\mathrm{e}}^{ W_{v}(t)}}\biggr\}= 1+\int_0^\infty \,{\mathrm{e}}^s\mathbb{P}\biggl\{{\sup_{t\in[0,T],v\in [0,1]}W_v(t)>s} \biggr\} \,{\mathrm{d}} s.\end{equation*}

By the Borell–TIS inequality and (3.5),

\begin{equation*}\mathbb{P}\biggl\{{\sup_{t\in[0,T], v\in [0,1]}W_v(t)>s} \biggr\}\leq {\exp\biggl({-\frac{(s-Q_1)^2}{2Q_2}}\biggr)}\end{equation*}

for sufficiently large s and $Q_1,Q_2$ positive constants, which justifies the limit above. Since by assumption (3.5), for some $C>0$ and all $z\in [0,1]$ , we have

\begin{equation*} \sigma_z^2(t) \le C (\lVert {t} \rVert^{\nu_{0}}+ \lVert {t} \rVert^{\nu_{\infty}}) \,=\!:\, \sigma^2(t),\end{equation*}

then for the Gaussian process

\begin{equation*}X(t)= W_0(t)+W_\infty(t)- \sigma^2(t)/2,\quad t\in [0,T],\end{equation*}

with $W_0,\,W_\infty$ mutually independent centered Gaussian processes with continuous sample paths, stationary increments and variance function $C\lVert {t} \rVert^{\nu_{0}},C\lVert {t} \rVert^{\nu_{\infty}}$ , respectively, applying [Reference Dębicki and Hashorva9, Theorem 3.1] we obtain for any $z\in [0,1]$

\begin{equation*} H_{Z_z}^\delta \le H_{\widetilde{Z}}^\delta \in (0,\infty ),\end{equation*}

where $\widetilde{Z}(t)=\exp(X(t))$ , $ t\in [0,T]$ , and

\begin{equation*} \mathbb{E}\biggl\{{ \sup_{t\in [0,T] \cap \delta \mathbb{Z}} \,{\mathrm{e}}^{ X_{z}(t)}}\biggr\} \le \mathbb{E}\biggl\{{ \sup_{t\in [0,T] \cap \delta \mathbb{Z}} \,{\mathrm{e}}^{ X(t)}}\biggr\}< \infty .\end{equation*}

Consequently, by the measurability of $A_T(z)$ , $z\in [0,1]$ , and the dominated convergence theorem,

\begin{equation*} \mathcal{H}_{{Z}}^\delta = \lim_{{T} \to \infty} T^{-d} \int_0^1 \mathbb{E}\biggl\{{\sup_{ t\in [0,T]^d \cap \delta \mathbb{Z}^d } \,{\mathrm{e}}^{ X_z (t)} } \biggr\} \,{\mathrm{d}} z = \int_0^1 H_{Z_{z}}^\delta \,{\mathrm{d}} z \le H^{\delta}_{\widetilde{Z}} < \infty .\end{equation*}

Thus, in view of Theorem 2.1, we obtain

\begin{equation*} \lim_{\delta \downarrow 0} \mathcal{H}_{{Z}}^\delta =\int_0^1\lim_{\delta \downarrow 0} H_{Z_z}^\delta \,{\mathrm{d}} z = \int_0^1 H_{Z_z}^0 \,{\mathrm{d}} z \ge 0,\end{equation*}

which completes the proof.