[Paper Review] On a General Approach to the Strong Laws of Large Numbers

István Fazekas|arXiv (Cornell University)|Jun 11, 2014

Probability and Risk Models39 references4 citations

TL;DR

This paper presents a general framework for deriving strong laws of large numbers (SLLN) using abstract Hájek-Rényi-type maximal inequalities, showing that such inequalities are consequences of underlying Kolmogorov-type inequalities. The method unifies and extends SLLN results for various dependent sequences—martingales, mixingales, associated sequences, and demimartingales—while enabling improved rates of convergence via a generalized Dini-type lemma.

ABSTRACT

A general method to obtain strong laws of large numbers is studied. The method is based on abstract Hájek-Rényi type maximal inequalities. The rate of convergence in the law of large numbers is also considered. Some applications for weakly dependent sequences are given.

Motivation & Objective

• To develop a unified theoretical framework for deriving strong laws of large numbers (SLLN) applicable to diverse dependent random sequences.
• To establish that Hájek-Rényi-type maximal inequalities follow systematically from Kolmogorov-type inequalities without assuming specific dependence structures.
• To extend existing SLLN results for weakly dependent processes, including martingales, associated sequences, and demimartingales, under a common abstract setting.
• To improve the rate of convergence in SLLN by introducing a generalized Dini-type lemma for moment inequalities.
• To provide a systematic method for deriving SLLN and rate-of-convergence results applicable across multiple stochastic processes without redundant proofs.

Proposed method

• Formalize abstract Kolmogorov-type and Hájek-Rényi-type maximal inequalities for moments, using non-negative sequences $\alpha_l$, $r>0$, and normalization sequences $\beta_l$.
• Prove that any Kolmogorov-type inequality with constant $K$ implies a corresponding Hájek-Rényi-type inequality with constant $C=4K$, independent of the normalization sequence $\beta_l$.
• Apply the abstract framework to derive SLLN by verifying the convergence condition $\sum_{l=1}^\infty \frac{\alpha_l}{b_l^r} < \infty$ for unbounded, non-decreasing sequences $b_n$.
• Use a generalized Dini-type lemma (Lemma 7.1) to derive improved rates of convergence, replacing classical Dini’s theorem in the analysis.
• Construct the rate function $\beta_n = \max_{1\leq k\leq n} b_k (\varphi(1/\nu_k))^{-1/r}$ with $\nu_k = \sum_{l=k}^\infty \frac{\alpha_l}{b_l^r}$, where $\varphi$ satisfies $\sum \frac{\varphi(n)}{n^2} < \infty$ and $\varphi(x) \uparrow \infty$.
• Demonstrate that the framework applies uniformly to various dependent processes, including $\varrho$-mixing, associated, AANA, and N-demimartingale sequences, by verifying the moment inequality condition.

Experimental results

Research questions

• RQ1 Can a single abstract framework unify the derivation of strong laws of large numbers across diverse dependent random sequences?
• RQ2 To what extent can Hájek-Rényi-type maximal inequalities be derived directly from Kolmogorov-type inequalities without additional assumptions?
• RQ3 How can the rate of convergence in the SLLN be systematically improved using generalized convergence criteria?
• RQ4 What is the role of the function $\varphi$ in refining the normalization sequence $\beta_n$ for optimal convergence rates?
• RQ5 In how many classes of dependent processes (e.g., martingales, demimartingales, associated sequences) does the abstract framework yield valid SLLN results?

Key findings

• The first Hájek-Rényi-type maximal inequality for moments follows from the first Kolmogorov-type inequality with a universal constant $C=4K$, independent of the normalization sequence $\beta_l$.
• The condition $\sum_{l=1}^\infty \frac{\alpha_l}{b_l^r} < \infty$ implies $\frac{S_n}{b_n} \to 0$ almost surely, providing a general criterion for SLLN under moment conditions.
• For sequences satisfying the Kolmogorov-type inequality with $\alpha_l$, $r>0$, and $K>0$, the normalized partial sums $S_n / \beta_n$ are almost surely bounded if $\{\beta_n\}$ is bounded, and converge to zero if unbounded.
• The rate of convergence is improved by replacing Dini’s theorem with Lemma 7.1, allowing the use of functions $\varphi$ satisfying $\sum \frac{\varphi(n)}{n^2} < \infty$ and $\varphi(x) \uparrow \infty$.
• The framework yields SLLN results for $\varrho$-mixing, associated, AANA, NOD, and demimartingale sequences by verifying the moment inequality condition.
• The normalization $\beta_n = \max_{1\leq k\leq n} b_k (\varphi(1/\nu_k))^{-1/r}$ ensures $\lim_{n\to\infty} \frac{\beta_n}{b_n} = 0$, implying stronger convergence than $b_n$ alone.