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

TL;DR

This paper establishes a central limit theorem for probability measures derived from the sum-of-digits function in base 2, showing that normalized sums tend to a normal distribution for almost all sequences.

Contribution

It introduces a novel approach to analyze sum-of-digits functions using matrix products and proves a CLT for measures associated with binary digit sums in a probabilistic setting.

Findings

Proves a CLT for measures based on sum-of-digits in binary.

Expresses probability measures as matrix products.

Demonstrates convergence to normal distribution for almost all sequences.

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).