Permutation-Invariance in Komlós' Theorem for Hilbert-Space Valued Random Variables

The Komlós theorem states that we can extract a subsequence from every $L_{\mathbb{R}}^{1}$-bounded sequence of random variables, so that every further subsequence converges Cesàro a.e. to the same limit. The purpose of this paper is to prove that if $\mathbb{H}$ is a Hilbert space, we can extract a subsequence from every $L_{\mathbb{H}}^{1}$-bounded sequence, so that every permuted subsequence converges Cesàro a.e. in $\mathbb{H}$ to the same limit.