On the Law of Large Numbers and Convergence Rates for the Random Projections

TL;DR

The paper studies random projections of a real-valued sequence $X_k$ by weights drawn uniformly from the unit sphere, and proves a Marcinkiewicz–Zygmund type law of large numbers under the weaker condition of finite $p$-th moments with $0<p<2$ via a mean-dominated tail by a variable $Y$. It connects random spherical weights to Gaussian-augmented representations, leverages truncation and gamma-function asymptotics to derive $L^p$ convergence and rate results, and establishes almost-sure convergence through series bounds. The results extend LLN-type convergence to non-identically distributed $X_k$ with explicit rates, including $n^{p/2-1}$-scale and $n^{1/2-1/p}$-scale bounds, and provide corollaries under stronger moment conditions. This advances understanding of randomized projections in high-dimensional settings and informs rates of convergence for estimators based on random projections under minimal moment assumptions.

Abstract

The aim of this paper is to establish the Marcinkiewicz-Zygmund (MZ) type law of large numbers for the randomly weighted sums with weights chosen randomly, uniformly over the unit sphere in $\mathbb{R}^n$. We also establish a theorem that describes the rate of convergence in the law of large numbers for these weighted sums.

Theorems & Definitions (13)

• Theorem 3.1
• Remark 3.1
• Theorem 3.2
• Corollary 3.1
• Theorem 3.3
• Corollary 3.2
• proof : Proof of Theorem \ref{['T1']}
• proof : Proof of Theorem \ref{['T3']}
• proof : Proof of Corollary \ref{['T3C1']}
• proof : Proof of Theorem \ref{['T4']}