Necessary and sufficient conditions for periodic decaying resolvents in linear discrete convolution Volterra equations and applications to ARCH$(\infty)$ processes

TL;DR

This paper establishes conditions under which solutions to linear discrete convolution Volterra equations exhibit periodic decaying behavior, and applies these results to analyze the autocovariance decay in ARCH(∞) processes.

Contribution

It provides necessary and sufficient conditions for periodic decaying resolvents in convolution equations and applies this to challenge existing assumptions about ARCH(∞) autocovariance decay.

Findings

Identified conditions linking kernel decay to solution decay.

Provided a counterexample to previous ARCH(∞) autocovariance decay results.

Analyzed specific examples illustrating the theory.

Abstract

We define a class of functions which have a known decay rate coupled with a periodic fluctuation. We identify conditions on the kernel of a linear summation convolution Volterra equation which give the equivalence of the kernel lying in this class of functions and the solution lying in this class of functions. Some specific examples are examined. In particular this theory is used to provide a counter--example to a result regarding the rate of decay of the auto--covariance function of an ARCH($\infty$) process.