Random walks on the circle and measure of irrationality

Bruno Massé (1)

1 LMPA - Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville

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.

Mots clés: Modulo 1 distribution; Benford law; Fourier coefficients; Weak convergence; Measure of irrationality

Domaines: Mathématiques [math]

Bruno Massé. Random walks on the circle and measure of irrationality. [Research Report] Université du Littoral Côte d'Opale. 2021, pp.1-14. ⟨hal-03274061v2⟩