On the stationary tail index of iterated random Lipschitz functions

Let ${\mathrm{\Psi }}_1,{\mathrm{\Psi }}_2,\mathrm{.}\mathrm{.}\mathrm{.}$ be a sequence of i.i.d. random Lipschitz functions on a complete separable metric space with unbounded metric $d$ and forward iterations $X_n$. Suppose that $X_n$ has a stationary distribution. We study the stationary tail behavior of the functional $D_n=d(x_0,X_n)$, $x_0$ an arbitrary reference point, by providing bounds for these random variables in terms of simple contractive iterated function systems on the nonnegative halfline. Our results provide bounds for the lower and upper tail index of $D_n$ and will be illustrated by a number of popular examples including the AR(1) model with ARCH errors and random logistic transforms.