Asymptotic distribution of Bernoulli quadratic forms

Abstract

Consider the random quadratic form ${\mathit{T}}_{\mathit{n}}={\sum }_{1\le \mathit{u}<\mathit{v}\le \mathit{n}}{\mathit{a}}_{\mathit{u}\mathit{v}}{\mathit{X}}_{\mathit{u}}{\mathit{X}}_{\mathit{v}}$, where ${(({\mathit{a}}_{\mathit{u}\mathit{v}}))}_{1\le \mathit{u},\mathit{v}\le \mathit{n}}$ is a $\{0,1\}$-valued symmetric matrix with zeros on the diagonal, and ${\mathit{X}}_1,{\mathit{X}}_2,\dots ,{\mathit{X}}_{\mathit{n}}$ are i.i.d. $Ber({\mathit{p}}_{\mathit{n}})$, with ${\mathit{p}}_{\mathit{n}}\in (0,1)$. In this paper, we prove various characterization theorems about the limiting distribution of ${\mathit{T}}_{\mathit{n}}$, in the sparse regime, where ${\mathit{p}}_{\mathit{n}}\to 0$ such that $\mathbb{E}({\mathit{T}}_{\mathit{n}})=\mathit{O}(1)$. The main result is a decomposition theorem showing that distributional limits of ${\mathit{T}}_{\mathit{n}}$ 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.