Asymptotic distribution of Bernoulli quadratic forms

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 diagonal, and X1,X2,…,Xn are i.i.d. Ber(pn), pn∈(0,1). In this paper, we prove various characterization theorems about limiting distribution of Tn, in sparse regime, pn→0 such that E(Tn)=O(1). The main result decomposition theorem showing distributional limits Tn sum three components: mixture which consists function independent Poisson variables; linear mixture, mean itself (possibly infinite) combination another component. This accompanied universality allows us to replace Bernoulli large class other discrete distributions. Another consequence general necessary sufficient condition for convergence, an interesting second moment phenomenon emerges.