Uniform Estimates of the Rate of Convergence in the Multi-Dimensional Central Limit Theorem

Abstract

The goal of this paper is to prove several estimates of the rate of convergence in the multi-dimensional central limit theorem in the uniform metric. Additionally, estimates will be derived for the rate of convergence in variation. These estimates are mainly derived from properties of the metrics. Formulation and Discussion of the ResultsConsider a sequence of independent identically distributed r.v.'s X1, X2, with values in Euclidean space R k. Let EX1 0, EIXIa< 0. Here[. [denotes the norm in R k. Below all the norms which we encounter will be denoted by I'1 should cause no misunderstandings. Denote by P and Pn the distributions of X1 and (X +. + Xn)n -1/2, respectively. We shall assume for simplicity that the covariance operator of the distribution P is the identity operator on R k. Denote by the normal law with zero mean and covariance operator tr2/, where ! is the identity operator in R k, and denote by . We shall use the following metrics: p(P, O)=sup{le(A)-O(A)l: A (}, where denotes the set of all convex Borel subsets of R k (we shall call p a uniform metric), as well as (P, O)= f IP-OI (dx), Var where s is the class of all real functions on R such that the integer rn and the number 0 < a =< 1 are such that m + a s, and f(') denotes the rnth FrOchet derivative of f, and by the norm of the derivative we mean the usual norm of a multilinear functional (see [2]-[5] for more details on the metrics (s).

Binary Time Series Modeling With Application to Adhesion Frequency Experiments

“…This is because the asymptotic distribution of the test is based on a K -dimensional central limit theorem. A necessary condition to maintain this asymptotic property is that K/N 1/5 → 0 (Senatov, 1980; Jiang, 2001b). Therefore, the proper number of partitions should be chosen from 1 to N 1/5 .…”

Sums of Independent Random Vectors: Proximity Estimating

“…Applying the techniques presented in Refs. [9,10] and Lemma 3.1 in Section 3, it is possible to show the existence constants c * , c * * that depend only on the distribution of the random vector X such that the condition…”

“…[9] for the particular case of normally distributed random vector . The above more general version is proved by the same method.…”

Normal approximation for strong demimartingales

“…Since the sequence {X * n , n ∈ N} is i.i.d., by Theorem 4 of Senatov (1980) and for a constant c 1 > 0 we have…”