Equidistribution of random walks on compact groups

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Abstract

Let X 1, X 2, . . . be independent, identically distributed random variables taking values from a compact metrizable group G. We prove that the random walk S k = X 1X 2 · · · X k, k = 1, 2, . . . equidistributes in any given Borel subset of G with probability 1 if and only if X 1 is not supported on any proper closed subgroup of G, and S k has an absolutely continuous component for some k ≥ 1. More generally, the sum ∑ k =1 N f (S k), where f : G → ℝ is Borel measurable, is shown to satisfy the strong law of large numbers and the law of the iterated logarithm. We also prove the central limit theorem with remainder term for the same sum, and construct an almost sure approximation of the process ∑ k≤ t f (S k) by a Wiener process provided S k converges to the Haar measure in the total variation metric.

Original languageEnglish
Pages (from-to)54-72
Number of pages19
JournalAnnales de l'Institut Henri Poincaré / Probabilités et Statistiques
Volume57
Issue number1
DOIs
Publication statusPublished - Feb 2021
Externally publishedYes

Keywords

  • Central limit theorem
  • Empirical distribution
  • Ergodic theorem
  • Law of the iterated logarithm
  • Strong law of large numbers
  • Wiener process

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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