Metric Distances in Spaces of Random Variables and Their Distributions

Золотарев, В. М.

doi:10.1070/sm1976v030n03abeh002280

Cited by 113 publications

(52 citation statements)

References 3 publications

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“…This lemma is a sharpening of the corresponding estimate in [3]. The improvement is that c(s) may be chosen so that it does not depend on the dimension.…”

Section: Auxiliary Resultsmentioning

confidence: 93%

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

Senatov¹

1981

Theory Probab. Appl.

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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).

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“…This lemma is a sharpening of the corresponding estimate in [3]. The improvement is that c(s) may be chosen so that it does not depend on the dimension.…”

Section: Auxiliary Resultsmentioning

confidence: 93%

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

Senatov¹

1981

Theory Probab. Appl.

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“…Ideal metrics of the given order s exist in the case of any Banach space U. That was shown in [7]. For reader's convenience we shall recall the definition given there.…”

Section: P Omentioning

confidence: 97%

“…In the solution of a number of problems of probability theory the method of metric distance functions has successfully been used lately. The essence of this method is based on the knowledge of the properties of metrics in spaces of random variables as well as on the principle according to which in every problem of the approximating type a metric as a comparison measure must be selected in accordance with the requirements to its properties (for a general acquaintance with the method we recommend papers [7,9] and [lOB. However, an example, illustrating the advantages of the method most expressively, is the use of the so-called ideal metrics in the problem of estimation of the remainder term in the central limit theorem.…”

Section: Introductionmentioning

confidence: 99%

Ideal Metrics in the Problems of Probability Theory and Mathematical Statistics

Золотарев¹

1979

Australian Journal of Statistics

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“…In addition, by the triangle R ( F ~, ~*)..< R (F*, F')+ t~ (F', r"' )+ K(,r," "" ~'). (8) In the sequel we shall systematically make use of the property of weak regularity of these metrics (see [8]): for any B~,~,~3~ ~ we have…”

Section: Ae~ Kmentioning

confidence: 99%

Approximation by infinitely divisible distributions in the multidimensional case

Зайцев¹

1984

J Math Sci

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Metric Distances in Spaces of Random Variables and Their Distributions

Cited by 113 publications

References 3 publications

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

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

Ideal Metrics in the Problems of Probability Theory and Mathematical Statistics

Approximation by infinitely divisible distributions in the multidimensional case

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