Theory of Probability and Mathematical Statistics
Smoothness and Lévy concentration function inequalities for distributions of random diagonal sums
Bero Roos
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Abstract: We present new explicit upper bounds for the smoothness of the distribution of the random diagonal sum Sn=Σj=1nXj,π(j) of a random nxn matrix X=(Xj,r), where are independent integer valued random variables, and π denotes a uniformly distributed random permutation on {1,...,n} independent of X. As a measure of smoothness, we consider the total variation distance between the distributions of Sn and 1+Sn. 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 Sn in the case of independent real valued random variables Xj,r.
Keywords: Generalized hafnian, Hoeffding permutation statistic, Lévy concentration function inequality, random diagonal sum, smoothness inequality
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