Exchangeable min-id sequences: Characterization, exponent measures and non-decreasing id-processes

Abstract

We establish a correspondence between exchangeable sequences of random variables whose finite-dimensional distributions are min-(or max-) infinitely divisible and non-negative, non-decreasing, infinitely divisible stochastic processes. The exponent measure of a min-id sequence is shown to be the sum of a very simple "drift measure" and a mixture of product probability measures, which corresponds uniquely to the Lévy measure of a non-decreasing infinitely divisible process. The latter is shown to be supported on non-negative and non-decreasing functions. Our results provide an analytic umbrella which embeds the de Finetti subfamilies of many classes of multivariate distributions, such as exogenous shock models, exponential and geometric laws with lack-of-memory property, minstable multivariate exponential and extreme-value distributions, as well as reciprocal Archimedean copulas with completely monotone generator and Archimedean copulas with log-completely monotone generator.

Exact simulation of continuous max-id processes

“…Assumption 1 requires that h X = 0, which is satisfied for every max-id sequence after suitable transformations of the margins. [2] describe the structure of exponent measures of exchangeable min-id sequences, i.e. of exchangeable sequences 1/X, where X is max-id.…”

“…This characterization is the analog to the usual Lévy-Khintchine triplet of infinitely-divisible distributions on R d [18]. The results of [2] imply that 1/X := (1/X i ) i∈N has the de Finetti type stochastic representation…”

“…copies of X. The key ingredient of their simulation algorithms is the PRM representation of X in (2). They exploit results from [5] about the conditional distribution of a specific decomposition of the PRM N to simulate only those functions which are relevant to determine the values of X at certain locations t 1 , .…”

“…This level of generality is necessary, since we want to put special emphasis on simulation algorithms for exchangeable max-id sequences which have locally compact (but not compact) index sets and possibly non-continuous marginal distributions. The focus on exchangeable max-id sequences is motivated by the recent results of [2], who uniquely characterize the distribution of exchangeable min-id sequences 1/X := (1/X i ) i∈N in terms of the law of a non-negative non-decreasing stochastic process H = (H t ) t∈R with the property that for every given n ∈ N there exist i.i.d. stochastic processes H (i,n) 1≤i≤n such that…”

“…Such processes are characterized by a so-called Lévy measure on the space of non-negative and nondecreasing functions and a non-negative and non-decreasing deterministic (drift-)function [17,2]. This characterization is the analog to the usual Lévy-Khintchine triplet of infinitely-divisible distributions on R d [18].…”