120883617
Berry-Esseen bounds for statistics of weakly dependent samples
We prove Berry-Esseen bounds for a general class of asymptotically normal statistics which are functions of N weakly dependent random variables under easily verifiable conditions. In particular, we show, for some 6 > 0, the validity of the bound o(N-'/~ log6 N) for U-statistics, studentized means, functions of sample means, functionals of empirical distribution functions and linear combinations of order statistics.
120655563
On the Rate of Convergence in the Central Limit Theorem in Banach Spaces
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 )
121633701
The Berry-Esseen bound for student's statistic
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.
122887101
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.
14271349
Uniform rates of convergence in the CLT for quadratic forms in multidimensional spaces
Summary. 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].
120814385
On Minimal Moment Assumptions in Berry–Esséen Theorems for U-Statistics
The rate of convergence for asymptotically normal U-statistics is of order $O(n^{ - {1 / 2}} )$ provided that \[ {\bf E}\left| {{\bf E}\left\{ {h\left( {X_1 ,X_2 } \right)\left| {X_1 } \right.} \right\}} \right|^3 0$. In this paper we extend the result for statistics of higher orders and with possible nonnormal limit distributions.
119807407
Über Teilerfunktionen in Dirichletschen Reihen