Max-stable processes

A stochastic process X ={X_t}_t∈T on an arbitrary index set T is said to be max-stable if for each n∈N and independent copies X⁽¹⁾, . . . , X⁽ⁿ⁾ of X the process of the maxima {^Wn_i=1X⁽ⁱ⁾}_t∈T has the same law as {a_n(t)Xt+bn(t)}_t∈T for suitable norming functions an(t)>0 and bn(t)∈Ron T.

Marginal distributions In particular, the univariate marginal distributions of X are max-stable. It is well-known that up to an affine transformation of the form x7→ax+bwitha >0 andb∈Rthe non-degenerate univariate max-stable distribu-tions are classified by belonging to one of the following three types (Fisher-Tippett theorem/Gnedenko’s theorem):

Φ_α(x) =

( 0 x≤0

exp (−x^−α) x >0 α >0 (Fréchet) Ψ_α(x) =

( exp (−(−x)^α) x <0

1 x≥0 α >0 (Weibull)

Λ(x) = exp −e^−x (Gumbel)

(cf. [31, 36]; see also [79, Proposition 0.3] or [16, Theorem 1.1.3] for the one-parameter representation due to von Mises and Jenkinson). Moreover, these marginal distributions can be transformed into each other by non-decreasing continuous trans-formations (cf. [23, p. 123]). Therefore and since we are interested in the dependence structure of max-stable processes, we shall primarily deal with max-stable processes X that havestandard Fréchet marginals as it is commonly done, i.e. X satisfies

P(Xt≤x) =

( 0 x≤0 e^−1/x x >0

fort∈T. Here the sequence of normalizing functions will bean(t) =nandbn(t) = 0 (cf. [23, p. 124]). Such standardized max-stable processes X will be called simple max-stable processes.

Finite-dimensional distributions In order to describe thefinite-dimensional distri-butions(f.d.d.) of a simple max-stable processX onT, we shall fix some convenient notation: LetM ⊂T be some non-empty finite subset ofT. ByR^M (resp. [0,∞]^M) we denote the space of real-valued (resp. [0,∞]-valued) functions on M. Elements of these spaces are denoted by x = (xt)_t∈M where xt = x(t). The standard scalar product is given through hx, yi = ^P_t∈Mx_ty_t. For a subset L ⊂ M we write 1_L for the indicator function of L in R^M (resp. [0,∞]^M), such that {1_{t}}_t∈M forms an orthonormal basis of R^M. In this sense, the origin of R^M equals 1_∅ being zero everywhere on M. Using this notation, we emphasize the fact that a multivariate distribution of a stochastic process is not any |M|-variate distribution, but it is bound to certain points (forming the set M) in the space T. Finally, we consider some reference norm k·k on R^M (not necessarily the one from the scalar product) and denote the positive unit sphere SM :={a∈[0,∞)^M : kak= 1}.

The f.d.d. of a finite sample {X_t}_t∈M of a simple max-stable process X may be described by means of one of the following three quantities that are all equivalent to the knowledge of the respective |M|-variate simple max-stable distribution of {X_t}_t∈M:

• the (finite-dimensional) spectral measure H_M (cf. [17] or [79, Proposition 5.11.]), i.e. the finite Radon measure onS_M such that forx∈[0,∞)^M \ {1_∅}

−logP(X_t≤x_t, t∈M) = Z

S_M

_

t∈M

a_t x_t

!

H_M(da), (1.1)

• the stable tail dependence function `_M (cf. [4, p. 257]), i.e. the function on

1.1. Max-stable processes 9

[0,∞)^M defined through

`_M(x) :=−logP(Xt≤1/xt, t∈M) = Z

SM

_

t∈M

at·xt

!

H_M(da), (1.2)

• the(finite-dimensional) dependency set K_M (cf. [66]), i.e. the largest compact convex setK_M ⊂[0,∞)^M satisfying

`_M(x) = sup{hx, yi : y∈ K_M} ∀x∈[0,∞)^M. (1.3) In order to be a valid finite-dimensional spectral measure of a simple max-stable random vector{X_t}_t∈M, the only constraint that a finite Radon measureH_M onS_M has to satisfy is that^R_S

M atHM(da) = 1 for allt∈M. This ensures standard Fréchet marginals. Moreover, up to this normalization to standard Fréchet marginals, it follows from [66] that stable tail dependence functions of multivariate simple max-stable distributions can be characterized as beingsublinear,homogeneous and max-completely alternating, whereas dependency sets are max-zonoids. We address this matter in more detail in Proposition A.5.1. Equation (1.3) expresses that`M is the support function ofK_M (cf. [87]).

Spectral representation Max-stable processes have a close connection to Poisson point processes. For theoretical background on Poisson point processes we refer to [13, 54, 79]. In [15] de Haan shows that all (simple) max-stable processes X = {X_t}_t∈T that are either defined on a countable index set T or defined on T = R and that are stochastically continuous may be represented as follows: There exists a finite measureνon the Borelσ-algebraB([0,1]) of [0,1] and non-negative measurable functions{V_t}_t∈T on [0,1] (with ^R₀¹Vt(ω)ν(dω) = 1 for each t∈T), such that

{X_t}_t∈T ^f.d.d.= (_∞

_

n=1

UnVt(ωn) )

t∈T

(1.4) in the sense of finite-dimensional distributions (f.d.d.), where{(U_n, ω_n)}^∞_n=1denotes an (enumerated) Poisson point process onR+×[0,1] with intensity u⁻²du×ν(dω).

The normalization ^R₀¹Vt(ω)ν(dω) = 1 is due to our choice of standard Fréchet marginals. For arbitrary unit Fréchet marginals with a different scale it is suffi-cient to require^R₀¹V_t(ω)ν(dω)<∞ instead.

In [92] Stoev and Taqqu introduce the slightly more general notion of anextremal stochastic integral by means of a random sup-measure M_ν with control measure ν, which allows to involvearbitrarycontrol measure spaces (Ω,A, ν) instead of

consid-ering ([0,1],B([0,1]), ν) as above. Indeed, the r.h.s. of (1.4) may be read as extremal stochastic integral

{X_t}_t∈T ^f.d.d.= Z e

Ω

V_t(ω)M_ν(dω)

t∈T

(1.5) with Ω = [0,1] and where M_ν denotes arandom sup-measure with control measure ν. We refer to [92] for a detailed explanation and to [96, p. 857] for an exploratory summary. For our purposes it will suffice to know that the f.d.d. of the process X from (1.5) are given by

−logP(X_t≤x_t, t∈M) = Z

Ω

_

t∈M

V_t(ω) x_t

!

ν(dω) (1.6)

forx∈[0,∞)^M \ {1_∅} and any non-empty finite subset M ⊂T.

Definition 1.1.1 (cf. [48, 96]). Let (Ω,A, ν) be a measure space and V ={V_t}t∈T

non-negative measurable functions (with ^R_ΩV_t(ω)ν(dω) = 1 for each t ∈ T). We call (Ω,A, ν, V) a spectral representation of the (simple) max-stable process X = {X_t}_t∈T, if (1.5) holds (or, equivalently, (1.6) holds for all non-empty subsets M ⊂ T). The functions {V_t}_t∈T and the measure ν will be called spectral functions and spectral measure, respectively. In case (Ω,A, ν) is a probability space, the collection of spectral functionsV ={V_t}_t∈T themselves form a stochastic process that will be addressed as spectral process.

Of course, any stochastic processXwith a spectral representation (1.5) is (simple) stable. Conversely, it has been shown in [48, Theorem 1] that all (simple) max-stable processes allow for a spectral representation on some sufficiently rich measure space (Ω,A, ν). Moreover, given a (simple) max-stable processXon a separable met-ric spaceT, the existence of a spectral representation (Ω,A, ν, V), where (Ω,A, ν) is a Lebesgue probability space (and the joint measurability of (t, ω)7→V_t(ω) in both variables t and ω) is guaranteed under mild conditions. This includes processes X that have a measurable modification and especially stochastically continuous pro-cessesX(cf. [48, Theorem 2], [96, Proposition 4.1.], [34, chapt. 3 sect. 3 Theorem 1]).

Such max-stable processesXare again representable in the form (1.4) (and not only (1.5)) with a spectral processV ={V_t}t∈T. It is convenient in this case to interpret the expression V(ω_n) in (1.4) as a sequenceV⁽ⁿ⁾ of independent copies of a process V = {V_t}_t∈T on T that are independent of the Poisson point process {U_n}^∞_n=1 on R+. However, we shall use other choices of (Ω,A, ν) when more appropriate for interpretations (as in the case of M3 processes, see Example 1.2.1).