Self-averaging sequences which fail to converge. (English) Zbl 1362.60034

The authors have studied sequences \(p(n)\) characterized by the property that each term is a weighted average over previous terms. In several examples in the literature, such sequences do not converge to a limit, which at first sight might be surprising. The main purpose of this paper is to demonstrate that it is natural to expect non-convergence if the largest weights in the average \(p(n)\) are given to values \(p(k)\) for which \(k\) is close to a fixed fraction of \(n\). It turns out that non-convergence is predictable or even inevitable under fairly weak conditions. The intuition is that fluctuations in \(p\) happen on a large scale, and if the averages are taken on a smaller scale, they cannot let the fluctuations vanish. The methods of the paper are illustrated by proving non-convergence for the group Russian roulette problem.


60F99 Limit theorems in probability theory
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