Tail probabilities for infinite series of regularly varying random vectors

Abstract

A random vector $X$ with representation $X={\sum }_{j\ge 0}{A}_j{Z}_j$ is considered. Here, $(Z_j)$ is a sequence of independent and identically distributed random vectors and $(A_j)$ is a sequence of random matrices, `predictable' with respect to the sequence $(Z_j)$. The distribution of $Z_1$ is assumed to be multivariate regular varying. Moment conditions on the matrices $(A_j)$ are determined under which the distribution of $X$ is regularly varying and, in fact, `inherits' its regular variation from that of the $(Z_j)$'s. We compute the associated limiting measure. Examples include linear processes, random coefficient linear processes such as stochastic recurrence equations, random sums and stochastic integrals.

Published in at http://dx.doi.org/10.3150/08-BEJ125 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

Cited by in corpus (7)

• Interplay of insurance and financial risks in a discrete-time model with strongly regular variation
• Aggregation of autoregressive random fields and anisotropic long-range dependence
• The tail process and tail measure of continuous time regularly varying stochastic processes
• Ruin probabilities under Sarmanov dependence structure
• On tail behaviour of stationary second-order Galton-Watson processes with immigration
• Moment conditions for random coefficient AR($\infty$) under non-negativity assumptions
• Tail asymptotics and precise large deviations for some Poisson cluster processes