Convergence of permuted products of exponentials

Abstract

Let $\{A_{i,n}\}$ be a triangular array of elements in a Banach algebra, whose norms do not grow too fast, and whose row averages converge to $A$. Let $\sigma \in S(n)$ be a permutation drawn uniformly at random. If the array only contains $o(n/\log n)$ distinct elements, then almost surely, for all $0<s<t<1$, the permuted product of their exponentials ${\prod }_{i=[sn]}^{[tn]}{e}^{A_{\sigma (i),n}/n}$ converges in norm to $e^{(t-s)A}$. For an array of finite-dimensional matrices, convergence holds without this restriction. The proof of the latter result consists of an estimate valid in a general Banach algebra, and an application of a matrix concentration inequality.