A central limit theorem and a moderate deviations principle for the ratio of geometric p-generalized arithmetic mean are shown. Also Berry–Esseen-type upper bound on rate convergence in is proved. This has implications probabilistic version question whether inequality between can be reversed or improved up to multiplicative constants. The involved random vectors we study belong class distributi...

Abstract Let $\mu $ be a probability measure on $\mathrm {GL}_d(\mathbb {R})$ , and denote by $S_n:= g_n \cdots g_1$ the associated random matrix product, where $g_j$ are i.i.d. with law . Under assumptions that has finite exponential moment generates proximal strongly irreducible semigroup, we prove Berry–Esseen bound optimal rate $O(1/\sqrt n)$ for coefficients of $S_n$ settling long-standing...

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...

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...

This paper studies the asymptotic distribution of descents $\des(w)$ in a permutation $w$, and its inverse, distributed according to Mallows measure. The measure is non-uniform probability on permutations introduced study ranked data. Under this measure, are weighted number inversions they contain, with weighting controlled by parameter $q$. main results Berry-Esseen theorem for $\des(w)+\des(w...

The distributions of the run occurrences for a sequence of independent and identically distributed (i.i.d.) experiments are usually obtained by combinatorial methods (see Balakrishnan and Koutras (2002, Chapter 5)) and the resulting formulae are often very tedious, while the distributions for non i.i.d. experiments are generally intractable. It is therefore of practical interest to find a suita...