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Regular variation of infinite series of processes with random coefficients

In this article, we consider a series $X(t)=\sum_{j \geq 1}Ψ_j(t) Z_j(t),t \in [0,1]$ of random processes with sample paths in the space $D=D[0,1]$ of càdlàg functions (i.e. right-continuous functions with left limits) on $[0,1]$. We assume that $(Z_j)_{j \geq 1}$ are i.i.d. processes with sample paths in $D$ and $(Ψ_j)_{j \geq 1}$ are processes with continuous sample paths. Using the notion of regular variation for $D$-valued random elements (introduced in Hult and Lindskog (2005)), we show that $X$ is regularly varying if $Z_1$ is regularly varying, $(Ψ_j)_{j \geq 1}$ satisfy some moment conditions, and a certain ``predictability assumption'' holds for the sequence $\{(Z_j,Ψ_j)\}_{j \geq 1}$. Our result can be viewed as an extension of Theorem 3.1 of Hult and Samorodnitsky (2008) from random vectors in $R^d$ to random elements in $D$. As a preliminary result, we prove a version of Breiman's lemma for $D$-valued random elements, which can be of independent interest.