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

Abstract: 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 $\mu$a such that, for any d, $\mu$a(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$\in$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).