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

We consider the three dimensional array $$\mathcal{A}$ = {a_{i,j,k}}{1\le i,j,k \le n}$,with$a{i,j,k} \in [0,1]$,andthetworandomstatistics$T_{1}:= \sum_{i=1}^n \sum_{j=1}^n a_{i,j,\sigma(i)}$and$T_{2}:= \sum_{i=1}^{n} a_{i,\sigma(i),\pi(i)}$,where$\sigma$and$\pi$ are chosen independently from the set of permutations of $\{1,2,\dots ,n\}\mathrm{.}$ These can be viewed as natural three dimensional generalizations of the statistic $T_3={\sum }_{i=1}^n{a}_{i,\sigma (i)}$, considered by Hoeffding \cite{Hoe51}. Here we give Bernstein type concentration inequalities for $T_1$ and $T_2$ by extending the argument for concentration of $T_3$ by Chatterjee \cite{Cha05}.