Tail probabilities for infinite series of regularly varying random vectors

Abstract

A random vector X with representation X = ∑ j≥0AjZj is considered. Here, (Zj) is a sequence of independent and identically distributed random vectors and (Aj) is a sequence of random matrices, 'predictable' with respect to the sequence (Zj). The distribution of Z1, is assumed to be multivariate regular varying. Moment conditions on the matrices (Aj) are determined under which the distribution of X is regularly varying and, in fact, 'inherits' its regular variation from that of the (Zj)'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. © 2008 ISI/BS.