Smoothness and L\'{e}vy concentration function inequalities for distributions of random diagonal sums

TL;DR

This paper derives explicit bounds on the smoothness and Levy concentration of the distribution of sums of random diagonal entries in matrices, using new inequalities related to matrix hafnians.

Contribution

It introduces new explicit bounds for distribution smoothness and Levy concentration for random diagonal sums, employing a novel inequality for generalized normalized matrix hafnians.

Findings

Bounds on total variation distance between S_n and 1+S_n

Upper bounds on Levy concentration function of S_n

Introduction of a new inequality for matrix hafnians

Abstract

We present new explicit upper bounds for the smoothness of the distribution of the random diagonal sum $S_n=\sum_{j=1}^nX_{j,\pi(j)}$ of a random $n\times n$ matrix $X=(X_{j,r})$, where the $X_{j,r}$ are independent integer valued random variables, and $\pi$ denotes a uniformly distributed random permutation on $\{1,\dots,n\}$ independent of $X$. As a measure of smoothness, we consider the total variation distance between the distributions of $S_n$ and $1+S_n$. Our approach uses a new auxiliary inequality for a generalized normalized matrix hafnian, which could be of independent interest. This approach is also used to prove upper bounds of the L\'{e}vy concentration function of $S_n$ in the case of independent real valued random variables $X_{j,r}$.