Equidistribution of random walks on compact groups

Publikation: Beitrag in einer FachzeitschriftArtikel

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.

Originalspracheenglisch
Seiten (von - bis)54-72
Seitenumfang19
FachzeitschriftAnnales de l'Institut Henri Poincaré / Probabilités et Statistiques
Jahrgang57
Ausgabenummer1
DOIs
PublikationsstatusVeröffentlicht - Feb 2021
Extern publiziertJa

ASJC Scopus subject areas

  • !!Statistics and Probability
  • !!Statistics, Probability and Uncertainty

Fingerprint

Untersuchen Sie die Forschungsthemen von „Equidistribution of random walks on compact groups“. Zusammen bilden sie einen einzigartigen Fingerprint.

Dieses zitieren