[Paper Review] Marcinkiewicz-Zygmund Strong Law of Large Numbers for Pairwise i.i.d. Random Variables

V. M. Korchevsky|arXiv (Cornell University)|Apr 27, 2014

Probability and Risk Models7 references3 citations

TL;DR

This paper establishes the Marcinkiewicz–Zygmund strong law of large numbers for pairwise i.i.d. random variables under the condition $\mathbb{E}|X_1|^p < \infty$ for $1 < p < 2$. By extending classical results from independent to pairwise independent sequences, it proves that $(S_n - \mathbb{E}S_n)/n^{1/p} \to 0$ almost surely, generalizing the classical theorem under weaker dependence assumptions using moment bounds and truncation techniques.

ABSTRACT

It is shown that the Marcinkiewicz-Zygmund strong law of large numbers holds for pairwise independent identically distributed random variables. It is proved that if $X_{1}, X_{2}, \ldots$ are pairwise independent identically distributed random variables such that $E|X_{1}|^p &lt; \infty$ for some $1 &lt; p &lt; 2$, then $(S_{n}-ES_{n})/n^{1/p} o 0$ a.s. where $S_{n} = \sum_{k=1}^{n} X_{k}$.

Motivation & Objective

• To extend the Marcinkiewicz–Zygmund strong law of large numbers from independent to pairwise independent identically distributed (i.i.d.) random variables.
• To determine whether the condition $\mathbb{E}|X_1|^p < \infty$ for $1 < p < 2$ implies almost sure convergence of $(S_n - \mathbb{E}S_n)/n^{1/p}$ under pairwise independence.
• To close a gap in the literature by showing that the classical Marcinkiewicz–Zygmund theorem holds under the weaker pairwise independence assumption.
• To provide a unified generalization of the strong law of large numbers for pairwise i.i.d. sequences by combining results from Etemadi, Sawyer, and Petrov with the current work.

Proposed method

• Truncating the random variables at level $n^{1/p}$ to control large deviations and applying moment inequalities.
• Using the Borel–Cantelli lemma to show that the truncated tails vanish almost surely, based on the integrability condition $\mathbb{E}|X_1|^p < \infty$.
• Applying moment bounds on truncated variables, particularly $\mathbb{E}(|X_i| \mathbb{I}_{\{|X_i| > n^{1/p}\}})$, and showing their normalized sum tends to zero.
• Employing a dyadic block argument, decomposing the sum $S_n$ into blocks of size $2^n$, and analyzing the behavior of the truncated partial sums within each block.
• Establishing almost sure convergence of the normalized centered sum by bounding the maximum deviation over each block using truncation and independence within blocks.
• Combining results from Lemmas 1–3 on tail behavior and moment decay with maximal inequality arguments to prove almost sure convergence of the normalized sum.

Experimental results

Research questions

• RQ1 Does the Marcinkiewicz–Zygmund strong law of large numbers hold for pairwise i.i.d. random variables when $\mathbb{E}|X_1|^p < \infty$ for $1 < p < 2$?
• RQ2 Can the independence assumption in the classical Marcinkiewicz–Zygmund theorem be weakened to pairwise independence without losing almost sure convergence?
• RQ3 What role does the integrability condition $\mathbb{E}|X_1|^p < \infty$ play in ensuring the convergence of $(S_n - \mathbb{E}S_n)/n^{1/p}$ under pairwise independence?
• RQ4 How do truncation and moment estimates on the tails of the distribution contribute to proving almost sure convergence in the pairwise independent case?

Key findings

• The Marcinkiewicz–Zygmund strong law of large numbers holds for pairwise i.i.d. random variables under the condition $\mathbb{E}|X_1|^p < \infty$ for $1 < p < 2$.
• The normalized centered sum $ (S_n - \mathbb{E}S_n)/n^{1/p} \to 0 $ almost surely, even when the random variables are only pairwise independent.
• The proof relies on truncating the random variables at $n^{1/p}$ and showing that the contribution of the tails vanishes almost surely via the Borel–Cantelli lemma.
• The expectation of the truncated variables satisfies $ \frac{1}{n^{1/p}} \sum_{i=1}^n \mathbb{E}[|X_i| \mathbb{I}_{\{|X_i| > n^{1/p}\}}] \to 0 $ as $n \to \infty$.
• The sum of variances of the truncated variables over dyadic blocks is finite, ensuring almost sure convergence of the block sums via maximal inequality arguments.
• The result generalizes Theorem A (Marcinkiewicz–Zygmund) to the pairwise i.i.d. case, completing a unified strong law for $0 < p < 2$ under pairwise independence.