Expectation of p-norm of random matrices with heavy tails

Abstract

The p-norm (p > 2) of a random matrix whose entries are gaussian, subgaussian and log concave have been studied previously. We conjecture the following generalization of the above results for heavy tailed random matrices: Conjecture 0.1. Let p > 2. Fix n > 0. A = (X[subscript ij]), i = 1,2,...,n,j = 1,2,...,N, be a random matrix whose entries are independent random variables with bounded 2p[superscript th] moments, where N ["greater than or slant equal to"] n[superscript p over 2]. Then, with high probability, ["for all"]x ["element of"] S[superscript n-1] : cN??[superscript /p] ["less than slant equal to"] ["double vertical bars"]Ax["double vertical bars"][subscript p] ["less than or slant equal to"] CN[superscript 1/p], for some constants, c,C > 0. We establish the following upper bound, generalising the work of [Latala05]: Theorem 0.2. Let p > 2. Fix n > 0. A = (X[subscript ij), i = 1,2,...,n,j = 1,2,...,N, be a random matrix whose entries are independent, identically distributed random variables with bounded 2p[superscript th] moments, where N ["greater than or slant equal to"] n[superscript p-1]. Then, with high probability, (close to 1):["for all"]x ["element of"] S[superscript n-1]: ["double vertical bars"]Ax["double vertical bars"][subscript p] ["less than or slant equal to"] CN[superscript 1/p] log N[log(logN)log[superscript 2](log(logN))][superscript 1-1/p] for some constant C, depending on p and the pth and 2pth moments of the random variable. We also get a similar upper bound when the entries of the random matrix are independent random variables with bounded 2pth moment and are not necessarily identically distributed.