Limiting Distributions of Sums with Random Spectral Weights

TL;DR

This paper investigates the asymptotic behavior of weighted sums involving i.i.d. variables and spectral weights, establishing central limit theorems under various conditions in the context of random matrix models.

Contribution

It introduces new central limit theorems for sums with spectral weights in the Erdős-Rényi-Gilbert model, expanding understanding of their asymptotic distributions.

Findings

Proves CLT for sums with spectral weights under certain conditions

Characterizes limiting distributions for weighted sums in random matrix models

Provides conditions for convergence to normal distribution

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 $X_1, X_2, \ldots, X_n$ are i.i.d.~random variables and $a_1, a_2, \ldots, a_n$ correspond to either eigenvalues or singular values in the classic Erd\H{o}s-R\'enyi-Gilbert model. In particular, we prove central limit-type theorems for the sequences $n^{-1}Z_n$ with varying conditions imposed on $X_1, X_2, \ldots, X_n$.