Functional limit theorems for divergent perpetuities in the contractive case

TL;DR

This paper studies the asymptotic behavior of divergent perpetuities where the sum diverges due to large additive terms, showing that their logarithm converges to an extremal process under regular variation assumptions.

Contribution

It establishes the weak convergence of the scaled logarithm of divergent perpetuities to an extremal process, extending understanding of their limit behavior in the contractive case.

Findings

Logarithm of divergent perpetuities converges to an extremal process.

Results apply to related Markov chains under regular variation.

Proofs use Skorokhod space and maximal function convergence.

Abstract

Let $\big(M_k, Q_k\big)_{k\in\mathbb{N}}$ be independent copies of an $\mathbb{R}^2$-valued random vector. It is known that if $Y_n:=Q_1+M_1Q_2+...+M_1\cdot...\cdot M_{n-1}Q_n$ converges a.s. to a random variable $Y$, then the law of $Y$ satisfies the stochastic fixed-point equation $Y \overset{d}{=} Q_1+M_1Y$, where $(Q_1, M_1)$ is independent of $Y$. In the present paper we consider the situation when $|Y_n|$ diverges to $\infty$ in probability because $|Q_1|$ takes large values with high probability, whereas the multiplicative random walk with steps $M_k$'s tends to zero a.s. Under a regular variation assumption we show that $\log |Y_n|$, properly scaled and normalized, converge weakly in the Skorokhod space equipped with the $J_1$-topology to an extremal process. A similar result also holds for the corresponding Markov chains. Proofs rely upon a deterministic result which establishes the $J_1$-convergence of certain sums to a maximal function and subsequent use of the Skorokhod representation theorem.