Asymptotic expansions in functional limit theorems
Asymptotic expansions for a class of functional limit theorems are investigated. It is shown that the expansions in this class fit into a common scheme, defined by a sequence of functions hn ([var epsilon]1,..., [var epsilon]n), n >= 1, of "weights" (for nobservations), which are smooth, symmetric, compatible and have vanishing first derivatives at zero. Then hn(n-1/2,..., n-1/2) admits an asymptotic expansion in powers of n-1/2. Applications to quadratic von Mises functionals, the C.L.T. in Banach spaces, and the invariance principle are discussed.

title: Asymptotic expansions for bivariate von Mises functionals
abstract: SummaryA Berry-Essen result and asymptotic expansions are derived for the distribution of bivariate von Mises functionals under moment and smoothness conditions.The results apply to the Cramér-von Mises ω2 — statistic as well as to the Central Limit Theorem in Hilbert space, yielding a convergence rate O(n−1+ɛ) for every ɛ>0 on centered ellipsoids.

title: On the Rate of Convergence in the Central Limit Theorem in Banach Spaces
abstract: On demontre un resultat et on l'utilise pour demontrer que le taux de convergence dans le theoreme limite fonctionnel pour des processus empiriques est d'ordre O )n −1/2 )

title: Asymptotic Distribution of Quadratic Forms
abstract: We consider quadratic forms Q n = Σ a jh X j X k , 1 ≤j ¬= k ≤ n where X j are i.i.d. random variables with finite third moment. We obtain optimal bounds for the Kolmogorov distance between the distribution of Q n and the distribution of the same quadratic forms with X j replaced by corresponding Gaussian random variables. These bounds are applied to Toeplitz and random matrices as well as to nonstationary AR(1) processes.

title: Uniform rates of convergence in the CLT for quadratic forms in multidimensional spaces
abstract: Let X,X1,X2,… be a sequence of i.i.d. random vectors taking values in a d-dimensional real linear space ℝd. Assume that EX=0 and that X is not concentrated in a proper subspace of ℝd. Let G denote a mean zero Gaussian random vector with the same covariance operator as that of X. We investigate the distributions of non-degenerate quadratic forms ℚ[SN] of the normalized sums SN=N−1/2(X1+⋯+XN) and show that
provided that d≥9 and the fourth moment of X exists. The bound ?(N−1) is optimal and improves, e.g., the well-known bound ?(N−d/(d+1)) due to Esseen (1945). The result extends to the case of random vectors taking values in a Hilbert space. Furthermore, we provide explicit bounds for ΔN and for the concentration function of the random variable ℚ[SN].

title: The Berry-Esseen bound for student's statistic
abstract: We prove the Berry-Esseen bound for the Student t-statistic. Under the assumption of a third moment this bound coincides (up to an absolute constant) with the classical Berry-Esseen bound for the mean. In general the distribution of the Student statistic converges to the standard normal distribution function at least as fast as the distribution of the mean, and sometimes faster. For example, rates of convergence can be proved if the underlying distribution is in the domain of attraction of the normal law.

title: Über Teilerfunktionen in Dirichletschen Reihen