On the stationary tail index of iterated random Lipschitz functions

TL;DR

This paper investigates the tail behavior of stationary distributions generated by iterated random Lipschitz functions, providing bounds on tail indices and illustrating with examples like AR(1) models and logistic transforms.

Contribution

It introduces bounds for the tail indices of stationary distributions in iterated Lipschitz systems, extending understanding of their tail behavior in complex stochastic models.

Findings

Bounds for lower and upper tail indices of $D_n$

Application to AR(1) with ARCH errors

Analysis of random logistic transforms

Abstract

Let $\Psi_1,\Psi_2,...$ 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.