Central limit theorem on the general linear group -

Abstract

Our goal in this thesis is to understand the Central Limit Theorem (CLT) for linear groups proved partially by Le Page in 1982 then fully by Benoist and Quint in 2016. Consider a probability measure m on the general linear group GL(d,R), with d ≥ 1, and (Yi)i a sequence of independent and identically distributed random variables on GL(d,R) of law m. We are interested in proving, under a natural moment condition on m and geometric assumptions on the semi-group generated by its support, that the sequence of random variables log∣∣Yn...Y₁∣∣, suitably normalized, converges to a Gaussian law. More precisely, assume that the semi-group generated by the support of m is strongly irreducible and contains a proximal element. Le Page proved in this context the CLT under an exponential moment of m and Benoist-Quint were able to weaken this assumption to the most natural one (in comparison to the case d=1): that of a moment of order 2.