%0 Journal Article
%T Central Limit Theorem for probability measures defined by sum-of-digits function in base 2
%+ Aix Marseille Université (AMU)
%+ Institut de Mathématiques de Marseille (I2M)
%A Emme, Jordan
%A Hubert, Pascal
%< avec comité de lecture
%J Annali della Scuola Normale Superiore di Pisa
%8 2017
%D 2017
%Z 1605.06297
%R 10.2422/2036-2145.201609_010
%Z Mathematics [math]/Probability [math.PR]
%Z Mathematics [math]/Number Theory [math.NT]Journal articles
%X 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).
%G English
%2 https://hal.science/hal-01318564v2/document
%2 https://hal.science/hal-01318564v2/file/TCL.pdf
%L hal-01318564
%U https://hal.science/hal-01318564
%~ CNRS
%~ UNIV-AMU
%~ EC-MARSEILLE
%~ I2M
%~ I2M-2014-