An approach to Sturm's law of large numbers for the Karcher mean of positive operators

Firstly we briefly review some available versions of the strong law of large numbers in Banach spaces and nonlinear extensions provided by Sturm in CAT(0) metric spaces. Sturm's 2001 result was directly applied to the case of the geometric (also called Karcher) mean of positive matrices, thus it lead to a natural formulation of the law for positive operators. However there are serious obstacles to overcome to prove the law in the infinite dimensional case. We propose to use a recently established gradient flow theory by Lim-P for the Karcher mean of positive operators and a proximal point approximation to prove the strong law of large numbers for the Karcher mean in the operator case, solving this problem.