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Random walks on the circle and measure of irrationality

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.
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https://hal.archives-ouvertes.fr/hal-03274061
Contributor : Bruno Massé Connect in order to contact the contributor
Submitted on : Saturday, December 25, 2021 - 11:24:15 AM
Last modification on : Friday, January 7, 2022 - 3:39:26 AM

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  • HAL Id : hal-03274061, version 2

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Bruno Massé. Random walks on the circle and measure of irrationality. [Research Report] Université du littoral côte d'Opale. 2021. ⟨hal-03274061v2⟩

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