Consider the random quadratic form , where is a -valued symmetric matrix with zeros on the diagonal, and are i.i.d. , with . In this paper, we prove various characterization theorems about the limiting distribution of , in the sparse regime, where such that . The main result is a decomposition theorem showing that distributional limits of 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.
The research of Sumit Mukherjee was partially supported by NSF Grant DMS-1712037.
The authors thank Shirshendu Ganguly for many illuminating discussions and Jordan Stoyanov for bringing to our attention the reference . The authors also thank the Associate Editor and the anonymous referees for their insightful comments, which greatly improved the presentation of the paper.
"Asymptotic distribution of Bernoulli quadratic forms." Ann. Appl. Probab. 31 (4) 1548 - 1597, August 2021. https://doi.org/10.1214/20-AAP1626