A leader-election procedure using records

TL;DR

This paper analyzes a stochastic leader-election process based on record values, establishing limit theorems for the number of rounds until all but the first person are eliminated, with applications to Poisson-Dirichlet coalescent collisions.

Contribution

It introduces a new analysis of a record-based leader-election algorithm, including limit theorems and normalization techniques involving tetrations, and connects these results to coalescent collision counts.

Findings

The number of rounds T(M) is tightly concentrated around log-star M.

Normalized labels converge to a non-Poissonian point process.

Limit theorems describe the asymptotic behavior of the process.

Abstract

The study of the number of collisions in a Poisson-Dirichlet coalescent leads to the analysis of the following version of a stochastic leader-elec\-tion algorithm. Consider an infinite family of persons, labeled by $1,2,3,\ldots$, who generate iid random numbers from an arbitrary continuous distribution. Those persons who have generated a record value, that is, a value larger than the values of all previous persons, stay in the game, all others must leave. The remaining persons are relabeled by $1,2,3,\ldots$ maintaining their order in the first round, and the election procedure is repeated independently from the past and indefinitely. We prove limit theorems for a number of relevant functionals for this procedure, notably the number of rounds $T(M)$ until all persons among $1,\ldots,M$, except the first one, have left (as $M\to\infty$). For example, we show that the sequence $(T(M)-\log^{*}M)_{M\in\mathbb{N}}$, where $\log^{*}$ denotes the iterated logarithm, is tight, and study its weak subsequential limits. We further provide an appropriate and apparently new kind of normalization (based on tetrations) such that the original labels of persons who stay in the game until round $n$ converge (as $n\to\infty$) to some random non-Poissonian point process and study its properties. The results are applied to study subsequential distributional limits for the number of collisions in the Poisson-Dirichlet coalescent.