Iterated Random Functions and Slowly Varying Tails

Abstract

Consider a sequence of i.i.d. random Lipschitz functions $\{{\Psi }_n{\}}_{n\ge 0}$. 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 ${\Psi }_0(t)\approx A_{0}t+B_0$. 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 $R_{n+1}=A_{n+1}{R}_n+B_{n+1}$..