Implicit renewal theory in the arithmetic case

TL;DR

This paper extends Goldie's implicit renewal theorem to the arithmetic case, revealing that the tail behavior of solutions to certain fixed point equations can be characterized by a logarithmically periodic function, showing significant differences from the nonarithmetic case.

Contribution

The paper develops an extension of Goldie's implicit renewal theorem to the arithmetic case, providing a detailed description of tail behavior involving logarithmic periodicity.

Findings

Tail of solution is of form $\,\ell (x) q(x) x^{-\,\kappa}$

Tail is not necessarily regularly varying

Uses renewal theoretic approach by Grincevičius and Goldie

Abstract

We extend Goldie's implicit renewal theorem to the arithmetic case, which allows us to determine the tail behavior of the solution of various random fixed point equations. It turns out that the arithmetic and nonarithmetic cases are very different. Under appropriate conditions we obtain that the tail of the solution $X$ of the fixed point equations $X \stackrel{\mathcal{D}}{=} AX + B$, $X \stackrel{\mathcal{D}}{=} AX \vee B$ is $\ell (x) q(x) x^{-\kappa}$, where $q$ is a logarithmically periodic function $q(x e^h) = q(x)$, $x > 0$, with $h$ being the span of the arithmetic distribution of $\log A$, and $\ell$ is a slowly varying function. In particular, the tail is not necessarily regularly varying. We use the renewal theoretic approach developed by Grincevi\v{c}ius and Goldie.