A Bernstein type inequality for sums of selections from three dimensional arrays

Abstract

We consider the three dimensional array A={ai,j,k}1≤i,j,k≤n, with ai,j,k∈[0,1], and the two random statistics T1≔∑i=1n∑j=1nai,j,σ(i) and T2≔∑i=1nai,σ(i),π(i), where σ and π are chosen independently from the set of permutations of {1,2,…,n}. These can be viewed as natural three dimensional generalizations of the statistic T3=∑i=1nai,σ(i), considered by Hoeffding (1951). Here we give Bernstein type concentration inequalities for T1 and T2 by extending the argument for concentration of T3 by Chatterjee (2005).