Random subsequences of {αn} with asymptotically independent successive terms - Archive ouverte HAL Access content directly
Reports (Research Report) Year : 2021

Random subsequences of {αn} with asymptotically independent successive terms

Abstract

Fix an irrational number α and let Y n be the number of attempts needed to get the nth success in a non-stationary sequence of independent Bernoulli trials. It is known that the law of the fractional part of αY n converges weakly to the uniform distribution on [0, 1) , as n → +∞, when the probabilities of success decrease to 0 and sum to +∞. We provide sufficient conditions on the probabilities of success ensuring that the fractional parts of αY n and αY n+1 are asymptotically independent. We extend our results to any number of successive terms, compute upper bounds of the convergence rates depending on a measure of irrationality of α and on the probabilities of success and apply our results to discuss the mantissae of 2 Yn and 2 Y n+1 .
Fichier principal
Vignette du fichier
Asymptotic independence 2 copie.pdf (381.17 Ko) Télécharger le fichier
Origin : Files produced by the author(s)

Dates and versions

hal-03502439 , version 1 (25-12-2021)

Identifiers

  • HAL Id : hal-03502439 , version 1

Cite

Bruno Massé. Random subsequences of {αn} with asymptotically independent successive terms. [Research Report] Université du Littoral - Côte d'Opale. 2021. ⟨hal-03502439⟩
34 View
36 Download

Share

Gmail Facebook X LinkedIn More