Permutation invariant strong law of large numbers for exchangeable sequences

TL;DR

The paper proves a permutation-invariant strong law of large numbers for exchangeable sequences. It combines the standard SLLN for exchangeable sequences with de Finetti’s theorem and the Komlós-Berkés theorem to show that for any subsequence and any permutation, the Cesàro averages converge almost surely to $\xi = \mathbb{E}[\xi_1|\mathscr{E}] = \mathbb{E}[\xi_1|\mathscr{T}]$, and that this limit equals the conditional expectations with respect to the exchangeable and tail sigma-algebras. This extends Komlós-type results to the whole sequence without subsequencing and clarifies how permutation invariance interacts with the conditional structure of exchangeable data. The findings have implications for analyzing sums of exchangeable observations under arbitrary reordering and subsequencing.

Abstract

We provide a permutation invariant version of the strong law of large numbers for exchangeable sequences of random variables. The proof consists of a combination of the Komlós-Berkes theorem, the usual strong law of large numbers for exchangeable sequences and de Finetti's theorem.

Theorems & Definitions (20)

• Proposition 1.1
• Theorem 1.2
• Remark 2.1
• Lemma 2.2
• proof
• Theorem 2.3: Komlós-Berkes theorem
• proof
• Theorem 2.4: De Finetti's theorem
• proof
• proof : Proof of Theorem \ref{['thm-main']}