A simple proof of heavy tail estimates for affine type Lipschitz recursions

TL;DR

This paper provides a straightforward proof that the tail of the stationary solution to affine Lipschitz recursions is regularly varying with a positive constant, simplifying previous complex arguments especially under Kesten-Goldie conditions.

Contribution

It offers a simple proof establishing the positivity of the tail constant for affine Lipschitz recursions, including cases satisfying Kesten-Goldie assumptions.

Findings

Proves the tail constant C is positive for affine recursions.

Simplifies the proof of heavy tail estimates in Lipschitz recursions.

Applicable to cases satisfying Kesten-Goldie conditions.

Abstract

We study the affine recursion $X_n = A_nX_{n-1}+B_n$ where $(A_n,B_n)\in {\mathbb R}^+ \times {\mathbb R} $ is an i.i.d. sequence and recursions $X_n = \Phi_n(X_{n-1})$ defined by Lipschitz transformations such that $\Phi (x)\geq Ax+B$. It is known that under appropriate hypotheses the stationary solution $X$ has regularly varying tail, i.e. $$\lim_{t\to\infty} t^{\alpha} {\mathbb P}[X>t] = C. $$ However positivity of $C$ in general is either unknown or requires some additional involved arguments. In this paper we give a simple proof that $C>0$. This applies, in particular, to the case when Kesten-Goldie assumptions are satisfied.