Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps

We consider the Markov chain $\{X_n^x{\}}_{n=0}^{\mathrm{\infty }}$ on ${\mathbb{R}}^d$ defined by the stochastic recursion $X_n^x={\text{p}}_{{\theta }_n}(X_{n-1}^x)$, starting at $x\in {\mathbb{R}}^d$, where ${\theta }_1,{\theta }_2,\mathrm{.}\mathrm{.}\mathrm{.}$ are i.i.d. random variables taking their values in a metric space $(\mathrm{\Theta },\mathfrak{r}),$ and ${\text{p}}_{{\theta }_n}:{\mathbb{R}}^d\mapsto {\mathbb{R}}^d$ are Lipschitz maps. Assume that the Markov chain has a unique stationary measure $\nu$. Under appropriate assumptions on ${\text{p}}_{{\theta }_n}$, we will show that the measure $\nu$ has a heavy tail with the exponent $\alpha >0$ i.e. $\nu (\{x\in {\mathbb{R}}^d:\mathrm{\mid }x\mathrm{\mid }>t\})\asymp t^{-\alpha }$. Using this result we show that properly normalized Birkhoff sums $S_n^x=\sum_{k=1}^n X_k^x$,convergeinlawtoan$\alpha$--stablelawfor$\alpha\in(0, 2]$.