On Extremal Index of max-stable stationary processes

On Extremal Index of max-stable stationary processes

24.11.2016
24.11.2016
Abstract

In this contribution we discuss the relation between Pickands-type constants defined for certain Brown-Resnick stationary process as

(set if ) and the extremal index of the associated max-stable stationary process . We derive several new formulas and obtain lower bounds for if is a Gaussian or a Lévy process. As a by-product we show an interesting relation between Pickands constants and lower tail probabilities for fractional Brownian motions.

Extremal index, mean cluster index, Pickands constant, M3 representation, Brown-Resnick stationary, max-stable process, Gaussian process, Lévy process.
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thankful for all the support and ideas you shared with us!\ShortTitleOn Extremal Index of Max-stable Stationary Processes \ShortAuthorsK. Dȩbicki and E. Hashorva\NumberOfAuthors2\FirstAuthorKrzysztof Dȩbicki\FirstAuthorCityWrocław\FirstAuthorAffiliationMathematical Institute, University of Wrocław\FirstAuthorAddresspl. Grunwaldzki 2/4,
50-384 Wrocław, Poland\FirstAuthorEmailKrzysztof.Debicki@math.uni.wroc.pl \FirstAuthorThanks \SecondAuthorEnkelejd Hashorva \SecondAuthorCityLausanne \SecondAuthorAffiliationUniversity of Lausanne\SecondAuthorAddressBâtiment Extranef, UNIL-Dorigny,
1015 Lausanne, Switzerland\SecondAuthorEmailEnkelejd.Hashorva@unil.ch\SecondAuthorThanks\ThirdAuthor\ThirdAuthorCity\ThirdAuthorAffiliation\ThirdAuthorAddress\ThirdAuthorEmail \ThirdAuthorThanks\FourthAuthor\FourthAuthorCity\FourthAuthorAffiliation\FourthAuthorAddress\FourthAuthorEmail\FourthAuthorThanks\FifthAuthor \FifthAuthorCity \FifthAuthorAffiliation\FifthAuthorAddress \FifthAuthorEmail\FifthAuthorThanks \MSCcodesPrimary: 60G15; Secondary: 60G70.

1 Introduction

The motivation for this contribution comes from the importance and the intriguing properties of the classical Pickands constants , which are defined for any by (interpret as

(1)

where

with a standard fractional Brownian motion with Hurst index , that is a centered Gaussian process with stationary increments and variance function .

It is well-known (but not trivial to prove) that is finite and positive for any . The only values known for are for and , see e.g., [41, 42]. Suprisingly, Pickands and related constant appear in numerous unrelated asymptotic problems, see e.g., the recent papers [17, 25, 26, 15].
The contribution [19] derived a new formula for Pickands constants, which in fact indicates a direct connection between those contants and max-stable stationary processes, see [11]. The definition of in \eqrefdefP is extended in [11] for some general process , provided that it defines a max-stable and stationary process. More precisely, assume throughout in the sequel that

(2)

where is a random process on the space of càdlàg functions
with

(3)

Hence satisfies almost surely, and .
If is a Poisson point process (PPP) with intensity on , and are independent copies of the random process being independent of , then the random process defined by

(4)

has unit Fréchet marginals and is max-stable. Here denotes the unit Dirac measure at .
Adopting the definition in [30], we shall refer to as the Brown-Resnick stationary process whenever the associated max-stable process is stationary. Note that stationarity of means that and have the same distribution for any .

In the sequel, for the case we shall assume that

for any compact . A direct consequence of stationarity of and the fact that for any and , see e.g., [18, 39]

(5)

is that, for any we have

Consequently, defined in \eqrefdefP exists and is given by (see [11])

(6)

Note that if , then \eqreffino implies that

for any , hence letting tend to 0 yields .

Interestingly, is related to the extremal index of the stationary process

where we set if . Indeed, by (5)

(7)

Thus the Fréchet limit result in \eqrefFreLim, which is already shown in [50] (see also [10][Proposition 3.1] and [18]) states that the extremal index of the stationary process is given for any by

(8)

Clearly, the constant is positive if and only if the extremal index of the stationary process is positive.

Numerous papers in the literature have discussed the calculation and estimation of extremal index of stationary processes, see e.g., the recent articles [46, 10, 38, 35, 33, 21] and the references therein.
The primary goal of this contribution is to study Pickands type constants by exploring the properties of the extremal index . In particular, we are interested in establishing tractable conditions that guarantee the positivity of .
By our assumptions it is clear that is stationary and jointly regularly varying, hence in view of [5, Theorem 2.1] (see also [29]), there exists the so-called tail process

of the stationary process , which was introduced in [5]. It turns out that for any we have the following stochastic representation

(9)

with a unit Pareto random variable with survival function being independent of the process .
Under the finite mean cluster size condition (see below Condition 2.1) and condition , see [5, 4, 32], it follows that is positive, see the seminal contribution [5].
We shall show the positivity of the extremal index under a weaker condition, namely supposing that

(10)

holds almost surely for . In our derivations the next simple result is crucial: {lemma} If are positive integers such that

then for any we have

(11)

In the next section we shall show that the new expression for the extremal index in \eqrefcandidat is positive under \eqrefasC. Exploiting the explicit form of the tail process we shall derive several new interesting formulas for .
Brief outline of the rest of the paper: Section 2 displays our main results which establish the positivity of the Pickands-type constants and some new formulas. In Section 3 we shall discuss the connection with mixed moving maxima (M3) representation of Brown-Resnick processes. Then we derive some explicit lower bounds for in case that in \eqrefWs is a Gaussian or a Lévy process and then discuss the relation between and the mean cluster index. Further, we shall show that the classical Pickands constants are related to a small ball problem. All the proofs are relegated to Section 4.

2 Main Results

We keep the same setup as in the Introduction and denote additionally by a unit exponential random variable which is independent of everything else. According to [5] a candidate for the extremal index is given by the following formula

(12)

where is the tail process of , see [5]. As in the aforementioned paper we shall impose the finite mean cluster size condition of [5, Condition 4.1]:

Condition 2.1

Given , there exists a sequence of positive integers satisfying such that

(13)

holds for any .

In view of [5, Proposition 4.2] we have that follows from Condition 2.1. Our main result below establishes new formulas for .
Moreover, from the above mentioned reference, Condition 2.1 together with well-known conditions of Hsing and Davis implies that the candidate of extremal index is equal to the extremal index, i.e., . It is well-known that is implied by the strong mixing of . However, our results derived below do not require strong mixing, but just mixing of .

{theorem}

Let with as in \eqrefWs be such that \eqrefB0 holds and is max-stable and stationary. We have that \eqrefasC holds for if and only if Condition \eqrefC holds. Moreover, if \eqrefasC holds for , then

(14)
(15)
(16)

where and is a unit exponential random variable independent of .

{remark}

a) If for any negative integer , then

holds for any integer . Consequently, by \eqrefformulaAB we have

(17)
(18)

which has been shown in [19] for the case is a standard fractional Brownian motion. The assumption can be removed, see [27].
b) Above we assumed that has càdlàg sample paths in order to define . For the results of Theorem 2, this assumption is not needed.
c) In [11] it is shown that under the assumptions of Theorem 2 we have

(19)

According to \eqrefalbinB, for calculation of it suffices to know , i.e., only the values of for positive matter. This is not the case for the formula \eqrefseb. Both \eqrefseb and \eqrefalbinB are given in terms of expectations and not as limits, which is a great advantage for simulations. To this end, we mention that simulation of Pickands constants has been the topic of many contributions, see e.g., [9, 36, 19].
d) If is Brown-Resnick stationary, i.e., the associated max-stable process with is max-stable and stationary, then the time reversed process also defines a Brown-Resnick stationary processes. Moreover, for any

Consequently the formulas in Theorem 2 can be stated with instead of , for instance we have

(20)
(21)

e) If with an random variable with distribution and probability density function , by \eqrefalbinA we have

(22)
(23)

holds for any . Consequently, letting we obtain the well-known result

A canonical example for with representation \eqrefWs is the case when is a centered Gaussian process with stationary increments, continuous sample paths, and variance function . Then the max-stable process is stationary, see [40]. Using a direct argument, we establish in the next theorem the positivity of . {theorem} If

(24)

then .

Since \eqrefln8 implies \eqrefasC, see Corollary 2.4 in [34] or [30], then using for any we immediately establish the positivity of .
Indeed, the positivity of is crucial for the study of extremes of Gaussian processes. Condition \eqrefln8 can be easily checked, for instance if . Consequently, the classical Pickands constants are positive for any . This fact is highly non-trivial; after announced in Pickands’ pioneering work [41], correct proofs were obtained later by Pickands himself, and in [7, 43], see for instance Theorem B3 in [8]. We note in passing that under general conditions on the positivity of is established in [13].
Apart from the alternative proof for the positiveness of the original Pickands constants, Theorem 2 extends to non-Gaussian processes . For the above Gaussian setup, direct calculations show the positivity of under a slightly weaker condition than \eqrefln8.

3 Discussions & Extensions

3.1 Relation with lower tail probabilities

For the classical case of Piterbarg constants , i.e., for we show below that \eqrefformulaABC implies a nice relation with a small ball problem. {proposition} For any we have

The above result strongly relates to the self-similarity property of fractional Brownian motion. In case of a general Gaussian , we still have that is stationary if has stationary increments. However, fBm is the only centered Gaussian process with stationary increments being further self-similar. Hence, no obvious extensions of the above relation with lower tails can be derived for general .

3.2 Non-Gaussian

The classical Pickands constants are defined for with a standard fBm with Hurst index . The more general case where is substituted by a centered Gaussian process with stationary increments is discussed in details in [13].
Our setup clearly allows for any random process , not necessarily Gaussian, which is Brown-Resnick stationary. Along with the Gaussian case of , the Lévy one has also been dealt already in the literature. In view of [23, 49], if is a Lévy process such that

then , is Brown-Resnick stationary, i.e., is max-stable stationary with unit Gumbel marginals.
In [31] an important constant appears in the asymptotic analysis of the maximum of standardised increments of random walks, which in fact is the Pickands constant introduced here for as above. In [31][Lemma 5.16] a new formula for is derived, which is identical with our formula in \eqrefkabWang. Another instance of the Pickands constant given by formula \eqrefformulaAB is displayed in [44][Theorem 5.3]. With the notation of that theorem, we have for that

where ’s are iid with the same distribution as for some with uniformly distributed on being independent of which has some pdf symmetric around 0.

Pickands constants appear also in the context of semi-min-stable processes, see [51]. In view of the aforementioned paper, several results derived here for max-stable processes are extendable to semi-min-stable processes.

3.3 Finite Mean Cluster Size Condition

As noted in [45], Condition 2.1 is implied by the so-called short-lasting exceedance condition given below:

Condition 3.1

Given , there exists a sequence of integers satisfying such that

(25)

is valid for any .

This latter condition is a rephrasing of the so-called B condition, see e.g., [1, 12, 2], which was formulated by discretising the original Berman’s condition, see [6]. Condition 3.1 is weaker than the condition of Leadbetter as discussed in [22][Section 5.3.2].
Commonly, Condition \eqrefC assumed for is referred to as the anti-clustering condition, see e.g., [46, 47]. Clearly, the finite mean cluster size condition is stronger then the anti-clustering condition. The latter appears in various contexts related to extremes of stationary processes, see e.g., [3, 37, 46, 5, 47] and the references therein.

3.4 M3 Representation

Since we assume that is max-stable stationary with càdlàg sample paths and with representation \eqrefWs is such that satisfies \eqrefB0, then assuming the following almost sure convergence

(26)

as is equivalent with the fact that possesses a mixed moving maxima representation (for short M3), see [20, Theorem 3] and [52]. More specifically, under \eqreffino2 we have the equality of finite dimensional distributions {align} ξ_W(t) =d max_i≥1 P_i e^ F_i(t- T_i),  t∈\mathbbR between rhs and lhs in \eqrefM3, where the ’s are independent copies of a measurable càdlàg process satisfying

(27)

almost surely, and is a PPP in with intensity with

(28)

Moreover , the restriction of on possesses an M3 representation for any , see [11] for more details. Denote the corresponding constant in the intensity of this PPP by (and thus is just given in \eqrefcw).
In view of [11][Proposition 1], if admits an M3 representation as mentioned above, then

(29)

provided that \eqrefasC holds. Hence Theorem 2 presents new formulas for . Note in passing that \eqreffino3 has been shown in [40]. Therein it is proved that is given by the right-hand side of \eqrefformulaABC assuming further that with a centered Gaussian process with statioanry increments satisfying almost surely.
In view of [11][Theorem 1], if \eqrefasC holds, then we have

(30)

with and . Thus .
The representation of as an expectation of the ratio is crucial for its simulation. Such a representation has been initially shown in [19] for classical Pickands constants.

3.5 Lower Bounds

In Theorem 2 we present new formulas for , which in turn establish the positivity of and thus the positivity for the extremal index of . If only the positivity of is of primary interest, then the conditions of Theorem 2 can be relaxed. Next, we consider two important classes of processes for that is centered Gaussian processes with stationary increments and Lévy processes. Results for the Lévy case has been already given in [11].
For particular values of , we show that it is possible to derive a positive lower bound for and thus establishing the positivity of . Let .

{theorem}

i) Let , , where is a centered Gaussian processes with stationary increments and variance function such that . Then for any

(31)

ii) Let , , where is a Lévy process satisfying \eqrefLevi. Then for any

(32)
{remark}

a) It follows from i) of Theorem 3.5 that if for all and some , then

(33)

Since for any , then the above implies
b) If is a Lévy process as in Theorem 3.5, ii), then (see the proof in Section 3)

(34)

3.6 Case

Since \eqrefFreLim holds also for and , then the extremal index of the continuous process is

which is positive, provided that \eqrefasC holds. In the special case that we have that

(35)

hence for such and for any

(36)

Recall that we denote by the extremal index of . Using the terminology of [28] we refer to defined by (assuming that the limit exists)

as the mean cluster index of the process . Since for any and

then clearly .
We show next that if possesses an M3 representation, then is positive. {proposition} Suppose that is max-stable and stationary with . If possesses an M3 representation and exists, then

(37)

holds for any .

{remark}

a) In view of Theorems 2 and 3 in [11] we have for some general as in \eqrefWs, with being Gaussian or Lévy process

(38)

is valid for any . Consequently, under these conditions and the setup of Proposition 3.6

(39)

b) If , by \eqrefdefP2 and \eqrefalbinA for any

(40)

with a unit exponential random variable independent of . Expression \eqrefalbinAA of the classical Pickands constant was initially derived in [1] for some general , see also recent contribution [2]. In [28], Proposition 3 or the formula in [24][p.44] the classical Pickands constant is the limit of a cluster index.

4 Proofs

Proof of Lemma 1: Since , then by \eqrefFreLim and \eqrefFreLim2

For any we have

where the last equality follows from \eqrefbase. The assumption that and imply

(41)

Consequently,

hence the claim follows.

Proof of Theorem 2: We show first stochastic representation \eqrefstochRep. Recall that and for we set

By \eqrefbase, the fact that and the assumption that almost surely, for any positive and we have

where is a unit Pareto random variable with survival function independent of the process . Hence the claim in \eqrefstochRep follows by [5][Theorem 2.1 (ii)]. Next by the above derivations for any sequence of integers for any (recall almost surely) we have

where we used the fact that as in \eqrefcnD, the condition implies

and

Consequently,

where we used the assumption \eqrefasC. Hence Condition 2.1 holds.
In light of [5, Proposition 4.2] we have that Condition 2.1 implies \eqrefasC. Moreover, since

[5, Proposition 4.2] and Lemma 1 imply

Consequently,

(42)
(43)
(44)
(45)
(46)

where the second last expression follows from the monotone convergence theorem. In fact, the above claim readily follows also from [5][Remark 4.7]. Further from \eqrefthA1 we obtain

with a unit exponetial random variable independent of .
Next, \eqrefformulaAB follows from [45][Eq. (16)]. Since further we assume \eqrefWs, then \eqrefformulaAB implies

(47)

for any , establishing thus the proof.

Proof of Theorem 2: By our assumption for all large

Consequently, by \eqrefboundGA we have for all large and some