Inequality for Variance of Weighted Sum of Correlated Random Variables and WLLN

TL;DR

This paper derives new variance inequalities for weighted sums of correlated random variables using Cauchy-Schwarz and positive semidefinite matrix methods, and applies these results to Chebyshev's inequality and WLLN.

Contribution

It introduces novel variance bounds for correlated variables with general weights and provides a new proof approach, extending the weak law of large numbers.

Findings

Derived upper bounds for variance of weighted sums

Established variance inequalities for sums with general weights

Applied inequalities to Chebyshev's inequality and WLLN

Abstract

The upper bound inequality for variance of weighted sum of correlated random variables is derived according to Cauchy-Schwarz's inequality, while the weights are non-negative with sum of 1. We also give a novel proof with positive semidefinite matrix method. And the variance inequality of sum of correlated random variable with general weights is also obtained. Then, the variance inequalities are applied to the Chebyshev's inequality and sufficient condition of weak law of large numbers (WLLN) for sum of correlated random variables .