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

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 .