I am trying to describe the behavior of the iterates of a contraction mapping when noise is added. Given
- a real valued random variable $X_{0}$
- a sequence $\{Z_{n}\}_{n=0}^{\infty}$ of i.i.d real valued r.v's with mean $0$
- a contraction mapping $T : \mathbb{R} \rightarrow \mathbb{R}$ (for example $T(x) = ax + b$, for $|a| < 1$)
Define the random variables $$X_{n+1} = T(X_{n}) + Z_{n}$$ Is it true that the $\{X_{n}\}$ converge in distribution to a random variable with mean $x^{*}$, where $x^{*}$ is the fixed point of $T$? It's clear to me that it shouldn't converge in any stronger sense because of the $Z_{n}$, but I'm not really sure how to go about showing convergence in distribution, although I think it should be true..