Regular Variation of Infinite Series of Processes with Random Coefficients

Abstract

In this article, we consider a series X(t) = ∑_{j ⩾ 1}Ψ_j(t)Z_j(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 (Z_j)_{j ⩾ 1} are i.i.d. processes with sample paths in , and (Ψ_j)_{j ⩾ 1} are processes with continuous sample paths. Using the notion of regular variation for -valued random elements (introduced in Ref.^[13]), we show that X is regularly varying if Z₁ is regularly varying, (Ψ_j)_{j ⩾ 1} satisfy some moment conditions, and a certain “predictability assumption” holds for the sequence {(Z_j, Ψ_j)}_{j ⩾ 1}. Our result can be viewed as an extension of Theorem 3.1 of Ref.^[15] 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.

Keywords:

• Breiman’s Lemma
• Linear processes
• Regular variation
• Stochastic recurrence equation

Mathematics Subject Classification:

• Primary 60G52
• Secondary 60G17, 62M10