Self-averaging sequences which fail to converge

TL;DR

This paper investigates self-averaging sequences that are constructed as weighted averages of previous terms, revealing conditions under which these sequences fail to converge, with applications demonstrated through the group Russian roulette problem.

Contribution

It provides a probabilistic interpretation of non-converging self-averaging sequences and establishes weak conditions for their non-convergence.

Findings

Sequences based on fixed fractions of n tend not to converge.

Probabilistic methods can predict non-convergence in such sequences.

Application to the group Russian roulette problem illustrates the theory.

Abstract

We consider self-averaging sequences in which each term is a weighted average over previous terms. For several sequences of this kind it is known that they do not converge to a limit. These sequences share the property that $n$th term is mainly based on terms around a fixed fraction of $n$. We give a probabilistic interpretation to such sequences and give weak conditions under which it is natural to expect non-convergence. Our methods are illustrated by application to the group Russian roulette problem.