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

TL;DR

This paper investigates the law of large numbers and convergence rates for randomly weighted sums in high-dimensional spaces, providing theoretical insights into their probabilistic behavior.

Contribution

It introduces a Marcinkiewicz-Zygmund type law for randomly weighted sums with uniform weights on the sphere and characterizes their convergence rates.

Findings

Established a MZ-type law for random projections

Derived convergence rate theorems for weighted sums

Provides theoretical bounds for high-dimensional probabilistic sums

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.