Functional limit theorems for divergent perpetuities in the contractive case

Dariusz Buraczewski,Alexander Iksanov

doi:10.1214/ecp.v20-3915

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 $Y_n:=Q_1+M_1Q_2+\ldots+M_1\cdot\ldots\cdot M_{n-1}Q_n$ 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.

Highlights

Let Mk, Qk k∈N be independent copies of a random vector M, Q with arbitrary dependence of the components, and let X0 be a random variable which is independent of Mk, Qk k∈N
The purpose of the present paper is to prove functional limit theorems for the Markov chains (Xn) and for the divergent perpetuities (Yn) under the aforementioned assumptions
Summarizing we have proved that lim lim sup In(γ)

Summary

Introduction

Let Mk, Qk k∈N be independent copies of a random vector M, Q with arbitrary dependence of the components, and let X0 be a random variable which is independent of Mk, Qk k∈N. In the case that X0 = 0 a.s. it is seen that Xn has the same law as Yn for each fixed n

Functional limit theorems for divergent perpetuities

With these at hand we can proceed as follows

Since n n