ASYMPTOTIC DISTRIBUTION OF BERNOULLI QUADRATIC FORMS

Consider the random quadratic form T-n = Sigma(1 <= u<= n) a(uv)X(u)X(v), where ((a(uv)))1(<= u,v <= n) is a {0, 1}-valued symmetric matrix with zeros on the diagonal, and X-1, X-2,..., X-n are i.i.d. Ber(p(n)), with p(n) is an element of (0, 1). In this paper, we prove various characterization theorems about the limiting distribution of T-n, in the sparse regime, where p(n) -> 0 such that E(T-n) = 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.

Automated summary

This paper studies the limiting distribution of a random quadratic form in the sparse regime, presenting various characterization theorems. The main result shows that the distributional limits of the form consist of three components, including a quadratic function, a linear Poisson mixture, and an independent Poisson component. A universality result allows for the replacement of the Bernoulli distribution with other discrete distributions.