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On Strong Law of Large Numbers for pairwise independent random variables

Alina Akhmiarova, Alexander Veretennikov

TL;DR

This work extends the strong law of large numbers (SLLN) to a mixture of pairwise independent summands in which a possibly infinite-mean block $Y_n$ may be arbitrarily dependent, heredity to the sequence $Z_n$ constructed from $X_n$ and $Y_n$ via a deterministic indicator. Under a Cesàro-type uniform integrability (Cesàro-UI) condition on $X_n$ and a sublinear moment condition with tail bounds on $Y_n$, together with a sparsity constraint on the use of $Y_n$, the paper proves $\tfrac{1}{n}\sum_{k=1}^n Z_k \to 0$ almost surely. This generalizes Kolmogorov's SLLN and Etemadi's approach to the pairwise independent setting by allowing a large, possibly dependent and heavy-tailed portion. The proof employs an Etemadi-type lemma for the $X_n$ part and a Sawyer69 truncation technique for the $Y_n$ part, combined with Kronecker's lemma to transfer tail-sum bounds into almost sure convergence, offering a flexible framework for LLN in heterogeneous independence regimes.

Abstract

A new version of a Strong Law of Large Numbers is proposed in this note for pairwise independent random variables. The main goal is to relax the assumption on a finite expectation for each term.

On Strong Law of Large Numbers for pairwise independent random variables

TL;DR

This work extends the strong law of large numbers (SLLN) to a mixture of pairwise independent summands in which a possibly infinite-mean block may be arbitrarily dependent, heredity to the sequence constructed from and via a deterministic indicator. Under a Cesàro-type uniform integrability (Cesàro-UI) condition on and a sublinear moment condition with tail bounds on , together with a sparsity constraint on the use of , the paper proves almost surely. This generalizes Kolmogorov's SLLN and Etemadi's approach to the pairwise independent setting by allowing a large, possibly dependent and heavy-tailed portion. The proof employs an Etemadi-type lemma for the part and a Sawyer69 truncation technique for the part, combined with Kronecker's lemma to transfer tail-sum bounds into almost sure convergence, offering a flexible framework for LLN in heterogeneous independence regimes.

Abstract

A new version of a Strong Law of Large Numbers is proposed in this note for pairwise independent random variables. The main goal is to relax the assumption on a finite expectation for each term.
Paper Structure (4 sections, 3 theorems, 28 equations)

This paper contains 4 sections, 3 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. The setting
  3. SLLN for partially pairwise independent random variables
  4. Proof of Theorem \ref{['thm1']}

Key Result

Theorem 1

Under the assumptions (eq2-8) -- (3-A3) the following SLLN holds true:

Theorems & Definitions (6)

  • Theorem 1
  • Remark 1
  • Lemma 1: for $(X_n)$
  • proof
  • Lemma 2: for $(Y_n)$
  • proof