Central Limit Theorem for probability measures defined by sum-of-digits function in base 2

Jordan Emme (1, 2) , Pascal Hubert (2, 1)

1 AMU - Aix Marseille Université
2 I2M - Institut de Mathématiques de Marseille

In this paper we prove a central limit theorem for some probability measures defined as asymtotic densities of integer sets defined via sum-of-digit-function. To any integer a we can associate a measure on Z called µa such that, for any d, µa(d) is the asymptotic density of the set of integers n such that s_2(n + a) − s_2(n) = d where s_2(n) is the number of digits " 1 " in the binary expansion of n. We express this probability measure as a product of matrices. Then we take a sequence of integers (a_X(n)) n∈N via a balanced Bernoulli process. We prove that, for almost every sequence, and after renormalization by the typical variance, we have a central limit theorem by computing all the moments and proving that they converge towards the moments of the normal law N (0, 1).

Domaines: Probabilités [math.PR]; Théorie des nombres [math.NT]

Jordan Emme, Pascal Hubert. Central Limit Theorem for probability measures defined by sum-of-digits function in base 2. 2016. ⟨hal-01318564v1⟩