Abstract
We present new explicit upper bounds for the smoothness of the distribution of the random diagonal sum S n = ∑ j = 1 n X j , π ( j ) S_n=\sum _{j=1}^nX_{j,\pi (j)} of a random n × n n\times n matrix X = ( X j , r ) X=(X_{j,r}) , where X j , r X_{j,r} are independent integer valued random variables, and π \pi denotes a uniformly distributed random permutation on { 1 , … , n } \{1,\dots ,n\} independent of X X . As a measure of smoothness, we consider the total variation distance between the distributions of S n S_n and 1 + S n 1+S_n . Our approach uses new auxiliary inequalities for a generalized normalized matrix hafnian and for inverse moments of non-negative random variables, which could be of independent interest. This approach is also used to prove upper bounds of the Lévy concentration function of S n S_n in the case of independent real valued random variables X j , r X_{j,r} .
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