On the almost sure boundedness of norms of some empirical operators

Abstract

Let X 1, X 2,… be i.i.d. random variables and h be a symmetric measurable real function. We show that the norms of operators on l 2 n given by the matrix ( 1 n h(X i, X j) δ i≠j) 1 ≤ i,j ≤ n are a.s. bounded if and only if h is square integrable.