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Register a new userA Bernstein type inequality for sums of selections from three dimensional arrays
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We consider the three dimensional array $mathcal{A} = {a_{i,j,k}}_{1le i,j,k le n}$, with $a_{i,j,k} in [0,1]$, and the two random statistics $T_{1}:= sum_{i=1}^n sum_{j=1}^n a_{i,j,sigma(i)}$ 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 ${1,2,ldots,n }.$ 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}.
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