Invariance properties of random vectors and stochastic processes based on the zonoid concept

Abstract

Two integrable random vectors ξ and ξ * in R d are said to be zonoid equivalent if, for each u ∈ R d , the scalar products ξ, u and ξ * , u have the same first absolute moments. The paper analyses stochastic processes whose finite-dimensional distributions are zonoid equivalent with respect to time shift (zonoid stationarity) and permutation of its components (swap invariance). While the first concept is weaker than the stationarity, the second one is a weakening of the exchangeability property. It is shown that nonetheless the ergodic theorem holds for swap-invariant sequences and the limits are characterised.

Representations of max-stable processes via exponential tilting

“…Further, Z i 's are independent copies of Z being also independent of Π. See for more details the important contributions [3][4][5][6][7][8][9][10][11][12][13]. We shall refer to ζ Z as the associated max-stable process of Z; commonly Z is referred to as the spectral process.…”

“…Surprisingly, as shown below this fact holds for a general Y satisfying (2.1); see also its extension in Lemma 8.1 covering the case P{Z(h) = −∞} > 0. The claim of Lemma 2.1 is included in [8] and [25]; a direct proof is mentioned in [26] which is elaborated in [27][Lemma 1.1]. We present yet another proof in Section 7.…”

Generalized Pickands constants and stationary max-stable processes

“…We denote by ε x the unit Dirac measure at x ∈ R. The Brown-Resnick process ξ W is both max-stable and stationary Kabluchko, 2009Kabluchko, , 2011Molchanov and Stucki, 2013;Molchanov et al, 2014). The stationarity means that the processes {ξ W (t), t ∈ R} and {ξ W (t + h), t ∈ R} have the same distribution for any h ∈ R. Moreover, the process ξ W arises naturally as the limit of suitably normalized pointwise maxima of independent copies of stationary Gaussian processes (Kabluchko et al, 2009, Theorem 17).…”

Swap-invariant sequences and random measures

“…As stated in [5,Theorem 17] for each swap-invariant sequence, the mean converges almost surely to an integrable random variable. We now demonstrate that, if the ergodic limit is different from zero and if the convergence is in L 1 , the limit can be used to characterize swap-invariant sequences as scaled exchangeable sequences under another probability measure.…”

“…An important property of a swap-invariant sequence ξ is that n −1 n j=1 ξ j → X almost surely as n → ∞ for some random variable X (cf. [5,Theorem 17]). In contrast to the ergodic theorem for integrable exchangeable sequences (see for example [3,Theorem 10.6]), this convergence is not necessarily in L 1 .…”