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Original Articles

Regular Variation of Infinite Series of Processes with Random Coefficients

Pages 420-438 | Received 01 Feb 2014, Accepted 01 Jun 2014, Published online: 28 Jul 2014
 

Abstract

In this article, we consider a series X(t) = ∑j ⩾ 1Ψj(t)Zj(t), t ∈ [0, 1] of random processes with sample paths in the space of càdlàg functions (i.e., right-continuous functions with left limits) on [0, 1]. We assume that (Zj)j ⩾ 1 are i.i.d. processes with sample paths in , and (Ψj)j ⩾ 1 are processes with continuous sample paths. Using the notion of regular variation for -valued random elements (introduced in Ref.[Citation13]), we show that X is regularly varying if Z1 is regularly varying, (Ψj)j ⩾ 1 satisfy some moment conditions, and a certain “predictability assumption” holds for the sequence {(Zj, Ψj)}j ⩾ 1. Our result can be viewed as an extension of Theorem 3.1 of Ref.[Citation15] from random vectors in to random elements in . As a preliminary result, we prove a version of Breiman’s lemma for -valued random elements, which can be of independent interest.

Mathematics Subject Classification:

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