On Berman Functions

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z(t)= \exp \left( \sqrt{ 2} B_H(t)- \left|t \right|^{2H}\right) , t\in \mathbb {R}$$\end{document}Z(t)=exp2BH(t)-t2H,t∈R with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_H(t),t\in \mathbb {R}$$\end{document}BH(t),t∈R a standard fractional Brownian motion (fBm) with Hurst parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H \in (0,1]$$\end{document}H∈(0,1] and define for x non-negative the Berman function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {B}_{Z}(x)= \mathbb {E} \left\{ \frac{ \mathbb {I} \{ \epsilon _0(RZ) > x\}}{ \epsilon _0(RZ)}\right\} \in (0,\infty ), \end{aligned}$$\end{document}BZ(x)=EI{ϵ0(RZ)>x}ϵ0(RZ)∈(0,∞),where the random variable R independent of Z has survival function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/x,x\geqslant 1$$\end{document}1/x,x⩾1 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \epsilon _0(RZ) = \int _{\mathbb {R}} \mathbb {I}{\left\{ RZ(t)> 1\right\} }{dt} . \end{aligned}$$\end{document}ϵ0(RZ)=∫RIRZ(t)>1dt.In this paper we consider a general random field (rf) Z that is a spectral rf of some stationary max-stable rf X and derive the properties of the corresponding Berman functions. In particular, we show that Berman functions can be approximated by the corresponding discrete ones and derive interesting representations of those functions which are of interest for Monte Carlo simulations presented in this article.


Introduction
In the study of sojourns of rf's in a series of papers by S. Berman, see e.g., [1,2] a key random variable (rv) and a related constant appear.Specifically, let Zptq " expp ?2B H ptq ´|t| 2H q, t P R, with B H a fractional Brownian motion (fBm) with Hurst parameter H P p0, 1s, that is a centered Gaussian process with stationary increments, V arpB H ptqq " |t| 2H , t P R and continuous sample paths.In view of [2, Thm 3.3.1,Eq. (3.3.6)] the following rv (hereafter It¨u is the indicator function) plays a crucial role in the analysis of extremes of Gaussian processes.Throughout this paper R is a 1-Pareto rv (ln R is unit exponential) independent of any other random element.
The distribution function of 0 pRZq is known only for H P t1{2, 1u.For H " 1 as shown in [2, Eq.
The so-called Berman function defined for all x ě 0 (see [2, Eq. (3.0.An important property of the Berman function is that for x " 0 it equals the Pickands constant, see [2,Thm 10.5.1] i.e., B Z p0q " H Z , where H Z is the so called generalised Pickands constant This fact is crucial since B Z p0q is the first known expression of H Z in terms of an expectation, which is of particular usefulness for simulation purposes, see [3][4][5] for details on classical Pickands constants.
As shown in [6] for all x ě 0 1 T Bpr0, T s, xq, B Z pr0, T s, xq :" Motivated by the above definition, in this contribution we shall introduce the Berman functions for given δ ě 0 with respect to some non-negative rf Zptq, t P R d , d ě 1 with càdlàg sample paths (see e.g., [7,8] for the definition and properties of generalised càdlàg functions) such that Specifically, for given non-negative δ, x define B δ Z pxq :" lim Here λ 0 pdtq " λpdtq is the Lebesgue measure on R d , 0Z d " R d and λ δ pdtq{δ d is the counting measure on δZ d if δ ą 0. Hence B δ Z pxq, δ ą 0 is the discrete counterpart of B Z pxq and B 0 Z pxq " B Z pxq.In general, in order to be well-defined for the function B δ Z pxq, x ě 0 some further restriction on the rf Z are needed.A very tractable case for which we can utilise results from the theory of max-stable stationary rf's is when Z is the spectral rf of a stationary max-stable rf Xptq, t P R d , see (2.1) below.
An interesting special case is when ln Zptq is a Gaussian rf with trend equal the half of its variance function having further stationary increments.We shall show in Lemma 4.3 that for such Z the corresponding Berman function B Z pxq appears in the tail asymptotic of the sojourn of a related Gaussian rf.
Organisation of the rest of the paper.In Section 2 we first present in Theorem 2.1 a formula for Berman functions and then in Corollary 2.3 and Proposition 3.1 we show some continuity properties of those functions.In Theorem 2.5 and Lemma 2.4 we present two representations for Berman functions and discuss conditions for their positivity.Section 3 is dedicated to the approximation of Berman functions focusing on the Gaussian case.All the proofs are postponed to Section 4.

Main Results
Let the rf Zptq, t P R d be as above defined in the non-atomic complete probability space pΩ, F, Pq.Let further Xptq, t P R d be a max-stable stationary rf, which has spectral rf Z in its de Haan representation (see e.g., [10,11]) Here Γ i " ř i k"1 V k with V k , k ě 1 mutually independent unit exponential rv's being independent of tZ piq u 8 i"1 which are independent copies of Z.For simplicity we shall assume that the marginal distributions of the rf X are unit Fréchet (equal to e ´1{x , x ą 0) which in turn implies EtZptqu " 1 for all t P R d .Suppose further that for all T ą 0 and Z has almost surely sample paths on the space D of non-negative càdlàg functions f : R d Þ Ñ r0, 8q equipped with Skorohod's J 1 -topology.We shall denote by D " σpπ t , t P T 0 q the σ-field generated by the projection maps π t : π t f " f ptq, f P D with T 0 a countable dense subset of R d .In view of [12,Thm 6.9] with α " 1, L " B ´1, see also [13,Eq. (5.2)] the stationarity of X is equivalent with valid for every measurable functional F : D Ñ r0, 8s such that F pcf q " F pf q for all f P D, c ą 0.Here we use the standard notation B h Zp¨q " Zp¨´hq, h P R d .
We shall suppose next without loss of generality (see [14,Lem 7.1]) that Under the assumption that X is stationary B δ Z pxq is well-defined for all δ, x non-negative as we shall show below.We note first that, see e.g., [6,15] lim where H δ Z is the discrete counterpart of the classical Pickands constant H Z " H 0 Z .Hence for any x ą 0 we have In view of (2.4) we have that S 0 ą 0 almost surely.Since we do not consider the case δ ą 0 and δ " 0 simultaneously, we can assume that S δ ą 0 almost surely (we can construct a spectral rf Z for X that guarantees this, see [14,Lem 7.3]).
In view of [15,Cor 2 The next result states the existence and the positivity of Berman functions presenting further a tractable formula that is useful for simulations of those functions.
Recall that when δ " 0 we interpret δZ d as R d .We establish below the Berman representation (1.1) for the general setup of this paper.
Theorem 2.5.If PtS 0 " 8u ă 1, then for all δ, x non-negative Corollary 2.6.Under the conditions of Theorem 2.5 we have that 0 pY q has a continuous distribution if Z has almost surely continuous trajectories.Moreover, B δ Z pxq ą 0 for all x ě 0 such that Pt δ pY q ą xu ą 0.

Approximation of B δ
Z pxq and its behaviour for large x We show first that B Z " B 0 Z can be approximated by considering B δ Z pxq and letting δ Ó 0.
Proposition 3.1.For all x ě 0 we have that We note in passing that for x " 0 we retrieve the approximation for Pickands constants derived in [15].
An approximation of B δ Z pxq can be obtained by letting T Ñ 8 and calculating the limit of For such an approximation we shall discuss the rate of convergence to B δ Z pxq assuming further that V ptq is a continuous and strictly increasing function, and there exists α 0 P p0, 2s and A 0 P p0, 8q such that lim sup where ¨ is the Euclidean norm.
A2 There exists α 8 P p0, 2s such that lim inf t Ñ8 σ 2 V ptq t α8 ą 0. The following theorem constitutes the main finding of this section.Theorem 3.2.Under A1-A2 we have for all δ, x non-negative and λ P p0, 1q lim (i) For x " 0 the rate of convergence in (3.1) agrees with the findings in [18].
(ii) The range of the parameter λ P p0, 1q in Theorem 3.2 cannot be extended to λ ě 1.Indeed, following [19], for V ptq " ?2B 1 ptq, δ " 0, T ą x and d " 1 we have where ϕp¨q is the pdf of an N p0, 1q rv.Consequently, we have In the rest of this section we focus on d " 1 log-Gaussian case.In view of (3.2) for some finite positive constant C lnpB δ Z pxqq " ´Cσ 2 V pxq, x Ñ 8.The next result gives logarithmic bounds for B δ Z pxq as x Ñ 8 that supports this hypothesis.
(i) If we suppose additionally that σ 2 V is regularly varying at 8 with parameter α ą 0, then it follows from Proposition 3.4 that (ii) If follows from the proof of Proposition 3.4 that under A1-A2 .
A functional F : D Ñ r0, 8s is said to be shift-invariant if F pf p¨´hqq " F pf p¨qq for all h P R d .
l Below we interpret 8 ¨0 and 0{0 as 0. The next result is a minor extension of [22,Lem 2.7].Lemma 4.2.If PtS 0 ă 8u " 1, then for all measurable shift invariant functional F and all δ, x nonnegative xE " F pY {xq Proof of Lemma 4.2 For all measurable functional F : D Ñ r0, 8s and all x ą 0 xEtF pY qIpY phq ą xqu " EtF pxB h Y qIpxY p´hq ą 1qu (4.4) is valid for all h P R d with B h Y ptq " Y pt ´hq, h, t P R d .Note in passing that B h Y can be substituted by Y in the right-hand side of (4.4) if F is shift-invariant.The identity (4.4) is shown in [8].For the discrete setup it is shown initially in [13,23] and for case d " 1 in [22].
Next, if x P p0, 1s, since Y p0q " R ą 1 almost surely and by the assumption on the sample paths we have that Pt δ pY {xq ą 0u " 1, recall PtΘp0q " 1u " 1.By Lemma 4.1 PtM Y,δ P p1, 8qu " 1, hence for all x ą 1 we have further that M Y,δ ą x implies δ pY {xq ą 0. Consequently, in view of (4.1) δ pY {xq{ δ pY {xq is well defined on the event M Y,δ ą x, x ą 1 and also it is well-defined for any x P p0, 1s.
Recall that λ δ pdtq is the Lebesgue measure on R d if δ " 0 and the counting measure multiplied by δ d on δZ d if δ ą 0. Let us remark that for any shift-invariant functional F , the functional F pY {xqItM Y,δ ą maxpx, 1qu δ pY q δ pY {xq is shift-invariant for all h P R d if δ " 0 and any shift h P δZ d if δ ą 0. Thus applying the Fubini-Tonelli theorem twice and (4.4) with functional F ˚we obtain for all δ ě 0, x ą 0 δ pY q δ pY {xq hence the proof follows.l Proof of Theorem 2.1 Let δ ě 0 be fixed and consider for simplicity d " 1.By the assumption we have Etsup tPr0,T s Zptqu ă 8 for all T ą 0. Since we assume that Ptsup tPR Zptq ą 0u " 1, then PtS 0 ą 0u " 1.
Using the assumption we have PtS η ă 8u ą 0 for all η ě 0 and thus by (2.3) we obtain Further, assuming for simplicity that T is a positive integer we get where the last convergence follows from (4.5).The same way we show that L M,T Ñ 0 as M Ñ 8 establishing the proof.
We prove next the second claim.In view of [15, proof of Prop 2.1] almost surely for all δ, η P r0, 8q Θp0q, tS η pΘq ă 8u " tS δ pΘq ă 8u.We proceed next with the case δ " 0, the other case follows with the same argument where it is important that η " kδ for the shift transformation.Taking δ " 0, η ą 0 we have Since Z has almost surely continuous trajectories we have A s X A s 1 " H if 0 ă s ă s 1 and x ą 0. Thus there are countably many s ą 0 such that PtA s u ą 0 because if there were not countably many ones we would find countably many disjoint A s such that ř PtA s u " 8. Thus we get (4.8) for almost all s ą 0. The continuity at x " 0 follows from the right continuity of (4.7).l Proof of Lemma 2.4 Item (i): In view of (2.5) and substituting Θptq " Q δ ptq{S δ pΘq to (2.10) we get Since S δ pQ δ q " 1 the claim follows.
For this choice of b by (2.3) we have EtV ptqu " EtZptqItS 0 ă 8uu PtS 0 pΘq ă 8u " EtZp0qItS 0 ă 8uu PtS 0 pΘq ă 8u " 1 for all t P R. Clearly, Ptsup tPR d V ptq ą 0u " 1.In view of [21] V is the spectral rf of a stationary max-stable rf X ˚with càdlàg sample paths and moreover S 0 pV q " ş R d V ptqλpdtq ă 8 almost surely.In view of [15, The last equality follows from (recall M Y P p1, 8q almost surely) In view of (4.1) for all x non-negative such that Pt δ pY q ą xu ą 0 we have that B δ Z pxq P p0, 8q, hence the proof follows.
Assume now that PtS 0 ă 8u P p0, 1q.In view of Lemma 2.4 we have with V ptq " Zptq|S 0 ă 8, which is well-defined since PtS 0 ă 8u ą 0 by the assumption.Since S 0 pV q ă 8 almost surely and Y ˚ptq " Y ptq|S 0 pΘq ă 8, t P R by the proof above In view of [22, Lem 2.5, Cor 2.9] and [14, Thm 3.8] and the above ItS 0 pΘq ă 8u * (4.9) and thus δ pRΘq ă 8 implies S 0 ă 8 almost surely.Hence the proof is complete.l Proof of Corollary 2.6 In view of (2.3), the representation (2.15) and the finiteness of B 0 Z pxq for all x ě 0, the monotone convergence theorem yields for all x 0 ě 0 lim xÓx 0 E " Itx 0 ď 0 pY q ă xu 0 pY q * " E " It 0 pY q " x 0 u 0 pY q * " 0 consequently, since by our assumption Lemma 4.1 implies Pt 0 pY q P p0, 8qu " 1, then Pt 0 pY q " x 0 u " EtIt 0 pY q " x 0 uu " 0 follows establishing the claim.Pt δ pY q ą xu 2 Et δ pY qIt δ pY q ą xuu ě Pt δ pY q ą xu 2 Et δ pY qu establishing the proof of the lower bound (2.16).The proof of the upper bound follows from the fact that where F is the distribution of δ pY q.This completes the proof.l Proof of Proposition 3.1 Since B δ Z p0q is the generalised Pickands constant H δ Z , then the claim follows for x " 0 from [15].In view of (2.14) we can assume without loss of generality that PtS 0 ă 8u " 1.
Under this assumption, from the proof of Lemma 4.1 we have that Y ptq Ñ 0 almost surely as t Ñ 8.
Hence for some M sufficiently large Y ptq ă 1 almost surely for all t such that t ą M .Consequently, for all δ ě 0 δ pY q " ż δZ d Xr´M,M s d ItY ptq ą 1uλ δ pdtq.
Next, by Lemma 4.4, for sufficiently large T, u and some Const 0 ą 0 The upper bound for Σ 1 follows by a similar argument as used in the proof of [25,Lem 6.3], thus we explain only main steps of the argument.For a while, consider the following probability where in (4.16) we used Bonferroni inequality.
Using that q K Ă p K with the upper bound for Pt δ pY q ą xu as x Ñ 8 and then to apply Proposition 2.7.In order to simplify the notation, we consider only the case δ " 0. Let Zptq " V ptq ´σ2 V ptq{2, t P R with V a centered Gaussian process with stationary increments that satisfies A1-A2 and W an independent of V exponentially distributed rv with parameter 1.

Figure 2 . 2 V
Figure 2. The graphs of B Z pxq and lnpB Z pxqq σ 2 V px{2q as function of x and the lower bound of Pickands constant as a function of δ for integrated Ornstein-Uhlenbeck process.

X
u ptq ą u, max tPR j X u ptq ą u * derived in (4.11), we conclude that for each T sufficiently large and ε ą 0, lim inf uÑ8 P !ş r0,lnpuqs d IpX u ptq ą uqdt ą x)plnpuqq d Ψpuq ě B Z pr0, T s d , xq T d ´Const 4 1 T d ´T d´1`ε `T 2d exp ´´T α8{2 ¯`T 2d exp ´´T ε{2 ¯¯.(4.18)Thus, by statement (ii) of Lemma 4.3 combined with (4.15) and (4.18), in view of the fact that ε can take any value in p0, 1q, we arrive at lim T Ñ8 ˇˇˇB Z pxq ´BZ pr0, T s d , xq T d ˇˇˇT λ " 0 for all λ P p0, 1q establishing the proof.l Proof of Proposition 3.4 The idea of the proof is to analyze the asymptotic upper and lower bound of [16, " 0, then we retrieve the results of [15, Prop 2.1].(ii)As shown in[15]condition (2.6) holds in the particular case that Zptq ą 0, t P R d almost surely.By the definition, Θ has also càdlàg sample paths and since D is Polish, in view of[16, Lem p. 1276] we can assume that Θ is defined in the same probability space as Z. Recall that λ δ pdtq{δ d is the counting measure on δZ d if δ ą 0 and λ 0 is the Lebesgue measure on R d .Since we can Ete s δ pY q u ď 1 `spEt δ pY quq 1{2

Table 1 .
[4]compute Et δ pY qu, see Tab. 1.The graph of Et δ pY qu as a function of δ and the upper bound (2.17) with p " 1 for Berman constants as a function of x P r1, 10s are presented on Fig.1.Values of Et by the negative correlation of increments for H ă 0.5 we cannot trust the simulation for H close to 0 and we estimated Berman constant for H ě 0.4 (see the half width of 95% confidence interval in Tab.2).Let us note that the estimator (2.15) for x " 0 is different from the estimator of Pickands constant in[4].Compare our simulation for x " 0 with the results of[4]for Pickands constant.

Table 2 .
Estimation of B Z pxq for fractional Brownian motion B H and the half width of 95% confidence interval.Example 3.7.Let Xptq, t P R be a stationary Ornstein-Uhlenbeck process, i.e., a centered Gaussian process with zero mean and covariance EtXptqXpsqu " expp´|t ´s|q, s, t P R. Then the random process

Table 3 .
Estimation of B Z pxq for integrated Ornstein-Uhlenbeck process and the half width of 95% confidence interval..

Table 4 .
Estimation of B δ Z p0q for integrated Ornstein-Uhlenbeck process and the half width of 95% confidence interval.
proof of Prop 2.1] we have that tS δ pΘq ă 8u " tS 0 pΘq ă 8u almost surely for all δ ą 0. Consequently, we obtain for all δ ą 0 IpΘptq ą sqλ δ pdtq ą x ¯ItS δ pΘq ă 8, S 0 pΘq ă 8u with c " 1 if δ " 0 and c " δ d otherwise.Set below Q δ " Q{c and for simplicity omit the subscript below writing simply M Y instead of M Y,δ .Since Y ptq{M Y ď 1 almost surely for all t P δZ d and PtM Y P p1, 8qu " 1, in view of Lemma 2.4 we have using further the Fubini-Tonelli theorem and Lemma 4.2

l
Proof of Proposition 2.7 In order to prove (2.16) note first that for any non-negative rv U with df G and x ě 0 such that PtU ą xu ą 0 Proof of Theorem 3.2.Suppose that V ptq, t P R d is a centered Gaussian field with stationary increments and variance function σ 2V p¨q that satisfies A1-A2.Then, by stationarity of increments σ 2 V p¨q is negative definite, which combined with Schoenberg's theorem, implies that for each u ą 0 V ps ´tq ˙, s, t P R d is positive definite, and thus a valid covariance function of some centered stationary Gaussian rf X u ptq, t P R d , where s ´t is meant component-wise.The proof of Theorem 3.2 is based on the analysis of the asymptotics of sojourn time of X u ptq.Since the idea of the proof is the same for continuous and discrete scenario, in order to simplify notation, we consider next only the case δ " 0.Before we proceed to the proof of Theorem 3.2, we need the following lemmas, where Zptq " exp ´V ptq ´σ2V ptq 2 īs as in Remark 2.2, Item (iii).Lemma 4.3.For all T ą 0 and x ě 0 d IpX u ptq ą uqdt ą x ) Ψpuq " B Z pr0, T s d , xq.