A switch convergence for a small perturbation of a linear recurrence equation

TL;DR

This paper investigates how a small random perturbation affects the convergence behavior of a linear recurrence, revealing a sharp transition known as the cut-off phenomenon under certain conditions.

Contribution

It demonstrates the occurrence of a switch convergence or cut-off phenomenon in perturbed linear recurrences with roots inside the unit circle.

Findings

Convergence to the limiting distribution is exponential when roots are inside the unit circle.

A sharp transition in convergence, or cut-off, is proven under specific conditions.

The results extend understanding of stochastic behavior in linear recurrence systems.

Abstract

In this article we study a small random perturbation of a linear recurrence equation. If all the roots of its corresponding characteristic equation have modulus strictly less than one, the random linear recurrence goes exponentially fast to its limiting distribution in the total variation distance as time increases. By assuming that all the roots of its corresponding characteristic equation have modulus strictly less than one and some suitable conditions, we prove that this convergence happens as a switch-type, i.e., there is a sharp transition in the convergence to its limiting distribution. This fact is known as a cut-off phenomenon in the context of stochastic processes.