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August 2021 Asymptotic distribution of Bernoulli quadratic forms
Bhaswar B. Bhattacharya, Somabha Mukherjee, Sumit Mukherjee
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Ann. Appl. Probab. 31(4): 1548-1597 (August 2021). DOI: 10.1214/20-AAP1626

Abstract

Consider the random quadratic form Tn= 1u<vnauvXuXv, where ((auv))1u,vn is a {0,1}-valued symmetric matrix with zeros on the diagonal, and X1,X2,,Xn are i.i.d. Ber(pn), with pn(0,1). In this paper, we prove various characterization theorems about the limiting distribution of Tn, in the sparse regime, where pn0 such that E(Tn)=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.

Funding Statement

The research of Sumit Mukherjee was partially supported by NSF Grant DMS-1712037.

Acknowledgments

The authors thank Shirshendu Ganguly for many illuminating discussions and Jordan Stoyanov for bringing to our attention the reference [19]. The authors also thank the Associate Editor and the anonymous referees for their insightful comments, which greatly improved the presentation of the paper.

Citation

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Bhaswar B. Bhattacharya. Somabha Mukherjee. Sumit Mukherjee. "Asymptotic distribution of Bernoulli quadratic forms." Ann. Appl. Probab. 31 (4) 1548 - 1597, August 2021. https://doi.org/10.1214/20-AAP1626

Information

Received: 1 January 2020; Revised: 1 August 2020; Published: August 2021
First available in Project Euclid: 15 September 2021

Digital Object Identifier: 10.1214/20-AAP1626

Subjects:
Primary: 05D99 , 60C05 , 60F05
Secondary: 05C80 , 60H05

Keywords: combinatorial probability , limit theorems , moment phenomena , Poisson approximation , Random quadratic forms

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.31 • No. 4 • August 2021
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