A Constant on a Uniform Bound of a Combinatorial Central Limit Theorem

Abstract

Let n be a positive integer and Y(i, j), i, j = 1, ..., n, be random variables with finite fourth moments. Let π be a random permutation on {1, ..., n} which independent of Y(i, j)'s. In this paper, we use Stein's method and the technique from

Esseen type bounds of the remainder in a combinatorial CLT

“…They contain explicit constants in the inequalities. For non-degenerated Y ij , Esseen type inequalities were stated by Neammanee and Suntornchost (2005), Neammanee and Rattanawong (2009) and Chen and Fang (2012) (see also comments on p.2 of Chen and Fang (2012)).…”

On the probabilities of moderate deviations for combinatorial sums

“…Further, non-asymptotic Esseen type bounds have been derived for accuracy of normal approximation of distributions of combinatorial sums. Such results have been obtained in Bolthausen [6], von Bahr [7], Ho and Chen [8], Goldstein [9], Neammanee and Suntornchost [10], Neammanee and Rattanawong [11], Chen, Goldstein and Shao [12], Chen and Fang [13], Frolov [14,15], and in Frolov [16] for random combinatorial sums.…”

On combinatorial strong law of large numbers and rank statistics

“…One can easy derive sufficient conditions for the combinatorial CLT from Esseen inequalities which give bounds for the accuracy of the normal approximation of distributions of S n / √ B n . One can find such inequalities in von Bahr [1], Ho and Chen [2], Botlthausen [3], Goldstein [4], Neammanee and Suntornchost [5], Neammanee and Rattanawong [6], Chen and Fang [7] for X's with finite third moments. Earlier asymptotic results on combinatorial CLT may be found in references therein.…”