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### Massé, B (2021)

#### Random subsequences of αn with asymptotically independent successive terms

Research Report, Université du Littoral - Côte d'Opale.

ISSN/ISBN: Not available at this time. DOI: Not available at this time.

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 Y_n and 2 Y_n+1 .

Bibtex:
@techreport{, TITLE = {{Random subsequences of \{$\alpha$n\} with asymptotically independent successive terms}}, AUTHOR = {Mass{\'e}, Bruno}, URL = {https://hal.archives-ouvertes.fr/hal-03502439}, TYPE = {Research Report}, INSTITUTION = {{Universit{\'e} du Littoral - C{\^o}te d'Opale}}, YEAR = {2021}, MONTH = Dec, PDF = {https://hal.archives-ouvertes.fr/hal-03502439/file/Asymptotic%20independence%202%20copie.pdf}, HAL_ID = {hal-03502439}, HAL_VERSION = {v1}, }

Reference Type: Technical Report

Subject Area(s): Probability Theory