Iterated Random Functions and Slowly Varying Tails

Consider a sequence of i.i.d. random Lipschitz functions $\{Ψ_n\}_{n \geq 0}$. Using this sequence we can define a Markov chain via the recursive formula $R_{n+1} = Ψ_{n+1}(R_n)$. 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 $Ψ_0(t) \approx A_0t+B_0$. 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 $R_{n+1} = A_{n+1}R_n+B_{n+1}$..

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