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 thr...