Limiting Distributions of Sums with Random Spectral Weights

Abstract

This paper studies the asymptotic properties of weighted sums of the form $Z_n={\sum }_{i=1}^n{a}_i{X}_i$, in which $X_1,X_2,\dots ,X_n$ are i.i.d.~random variables and $a_1,a_2,\dots ,a_n$ correspond to either eigenvalues or singular values in the classic Erdős-Rényi-Gilbert model. In particular, we prove central limit-type theorems for the sequences $n^{-1}{Z}_n$ with varying conditions imposed on $X_1,X_2,\dots ,X_n$.