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Massé, B (2021)

Random walks on the circle and measure of irrationality

Research Report, Université du littoral côte d'Opale.

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

Abstract: Let Y_n be the number of attempts needed to get the nth success in a nonstationary sequence of independent Bernoulli trials and denote by α a fixed irrational number. We prove that, under mild conditions on the probabilities of success, the law of the fractional part of αY_n converges weakly to the uniform distribution on [0, 1) whenever α is irrational. We then compute upper bounds of the convergence rates depending on a measure of irrationality of α and on the probabilities of success. As an application, we discuss the mantissa of a Yn for positive integer a and the mantissa of the nth random Mersenne number generated by the Cramér model of pseudo-primes.

@techreport{, TITLE = {{Random walks on the circle and measure of irrationality}}, AUTHOR = {Mass{\'e}, Bruno}, URL = {}, TYPE = {Research Report}, INSTITUTION = {{Universit{\'e} du littoral c{\^o}te d'Opale}}, YEAR = {2021}, MONTH = Dec, PDF = {}, HAL_ID = {hal-03274061}, HAL_VERSION = {v2}, }

Reference Type: Technical Report

Subject Area(s): Probability Theory