University of Southern California and University of Melbourne We introduce a new family of distributions to approximate P(W ∈A) for A ⊂ {. . . ,−2,−1,0,1,2, . . .} and W a sum of independent integer-valued random variables ξ1, ξ2, . . . , ξn with finite second moments, where, with large probability, W is not concentrated on a lattice of span greater than 1. The well-known Berry–Esseen theorem s...

Recursive and closed form upper bounds are o¤ered for the Kolmogorov and the total variation distance between the standard normal distribution and the distribution of a standardized sum of n independent and identically distributed random variables. The approximation error in the CLT obtained from these new bounds vanishes at a rate O(n ); provided that the common distribution of the summands po...

Bounds in the mean central limit theorem, where the L1 distance is used to measure the discrepancy of the distribution Fn of a standardized sum of i.i.d. random variables with distributionG from the normal, is of some interest when the normal approximation is to be applied over some wide, and perhaps unspecified range of values. Esseen (1958) showed that the limiting value lim n→∞ n||Fn − Φ||1 ...

Refinements of first order asymptotic results are 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 U-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 e...