Asymptotic distribution of Bernoulli quadratic forms

Abstract

Consider the random quadratic form Tn=∑ 1≤u<v≤nauvXuXv, where ((auv))1≤u,v≤n is a {0,1}-valued symmetric matrix with zeros on the diagonal, and X1,X2,…,Xn are i.i.d. Ber(pn), with pn∈(0,1). In this paper, we prove various characterization theorems about the limiting distribution of Tn, in the sparse regime, where pn→0 such that E(Tn)=O(1). The main result is a decomposition theorem showing that distributional limits of Tn is the sum of three components: a mixture which consists of a quadratic function of independent Poisson variables; a linear Poisson mixture, where the mean of the mixture is itself a (possibly infinite) linear combination of independent Poisson random variables; and another independent Poisson component. This is accompanied with a universality result which allows us to replace the Bernoulli distribution with a large class of other discrete distributions. Another consequence of the general theorem is a necessary and sufficient condition for Poisson convergence, where an interesting second moment phenomenon emerges.