Let λ be a partition of n chosen from the Plancherel measure of the symmetric group Sn, let χλ(12) be the irreducible character of the symmetric group parameterized by λ evaluated on the transposition (12), and let dim(λ) be the dimension of the irreducible representation parameterized by λ. Fulman recently obtained the convergence rate of O(n−s) for any 0 < s < 1 2 in the central limit theorem...

An analogue of the Berry-Esseen inequality is proved for the speed of convergence of free additive convolutions of bounded probability measures. The obtained rate of convergence is of the order n, the same as in the classical case. An example with binomial measures shows that this estimate cannot be improved without imposing further restrictions on convolved measures. Courant Institute of Mathe...

Some limit theorems (including a Berry-Esseen bound) are derived for the number of comparisons taken by the Boyer-Moore algorithm for finding the occurrences of a given pattern in a random Markovian text. Previously, only special variants of this algorithm have been analyzed. We also propose means of computing the limiting constants for the mean and the variance.

Refinements of first order asymptotic results axe reviewed, with a number of Ph.D. projects supervised by van Zwet serving as stepping stones. Berry-Esseen bounds and Edgeworth expansions are discussed for R-, Land [/-statistics. After these special classes, the question about a general second order theory for asymptotically normal statistics is addressed. As a final topic, empirical Edgeworth ...

Let T be a general sampling statistic that can be written as a linear statistic plus an error term. Uniform and non-uniform Berry-Esseen type bounds for T are obtained. The bounds are best possible for many known statistics. Applications to U-statistic, multi-sample U-statistic, L-statistic, random sums, and functions of non-linear statistics are discussed.

We obtain rates of convergence and asymptotic expansions in limit theorems for powers of reciprocals of random variables. Our results improve on recent work of Shapiro [18], who obtained Berry-Esseen bounds of order n~ l/9+c (e > 0). Under weaker conditions than Shapiro imposed we obtain asymptotic expansions whose first term is of order n~ i (logn) 2 .

We derive uniform and non–uniform error bounds in the normal approximation under a general dependence assumption. Our method is tailor made for dynamic time series models employed in the econometric literature but it is also applicable for many other dependent processes. Neither stationarity nor any smoothness conditions of the underlying distributions are required. If the introduced weak depen...