Iterated Random Functions and Slowly Varying Tails

TL;DR

This paper investigates the tail behavior of the stationary distribution of a Markov chain generated by iterated random Lipschitz functions, revealing that under subexponential conditions, the tail can be characterized by the integrated tail of the logarithm of the coefficients.

Contribution

It provides new asymptotic tail results for the stationary distribution of recursive equations with Lipschitz functions under subexponential tail assumptions.

Findings

Tail asymptotics are described using the integrated tail function.

Results apply to the random difference equation R_{n+1} = A_{n+1} R_n + B_{n+1}.

Established conditions under which the tail behavior can be precisely characterized.

Abstract

Consider a sequence of i.i.d. random Lipschitz functions $\{\Psi_n\}_{n \geq 0}$. Using this sequence we can define a Markov chain via the recursive formula $R_{n+1} = \Psi_{n+1}(R_n)$. It is a well known fact that under some mild moment assumptions this Markov chain has a unique stationary distribution. We are interested in the tail behaviour of this distribution in the case when $\Psi_0(t) \approx A_0t+B_0$. We will show that under subexponential assumptions on the random variable $\log^+(A_0\vee B_0)$ the tail asymptotic in question can be described using the integrated tail function of $\log^+(A_0\vee B_0)$. In particular we will obtain new results for the random difference equation $R_{n+1} = A_{n+1}R_n+B_{n+1}$..