Approximation of Distributions of Sums of Independent Random Variables with Values in Infinite-Dimensional Spaces

Золотарев, В. М.

doi:10.1137/1121086

Cited by 73 publications

(59 citation statements)

References 10 publications

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“…is the identity operator in R k, and denote by . 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). 746 v.v.…”

Section: Formulation and Discussion Of The Resultsmentioning

confidence: 99%

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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Section: Formulation and Discussion Of The Resultsmentioning

confidence: 99%

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

Senatov¹

1981

Theory Probab. Appl.

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“…Both methods are technically more involved because we are dealing with recurrences with two parameters. We will indeed prove a stronger approximation to (5) by deriving a rate under the Zolotarev metric (see Zolotarev, 1976).…”

Section: Corollary 2 If 0 ˛ 1 Thenmentioning

confidence: 96%

Profiles of Random Trees: Limit Theorems for Random Recursive Trees and Binary Search Trees

2006

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We prove convergence in distribution for the profile (the number of nodes at each level), normalized by its mean, of random recursive trees when the limit ratio˛of the level and the logarithm of tree size lies in OE0; e/. Convergence of all moments is shown to hold only for˛2 OE0; 1 (with only convergence of finite moments when˛2 .1; e/). When the limit ratio is 0 or 1 for which the limit laws are both constant, we prove asymptotic normality for˛D 0 and a "quicksort type" limit law for˛D 1, the latter case having additionally a small range where there is no fixed limit law. Our tools are based on contraction method and method of moments. Similar phenomena also hold for other classes of trees; we apply our tools to binary search trees and give a complete characterization of the profile. The profiles of these random trees represent concrete examples for which the range of convergence in distribution differs from that of convergence of all moments.

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“…The Zolotarev metric has been studied in the context of distributional recurrences systematically in [14]. We collect the properties that are used subsequently, which can be found in Zolotarev [17,18] …”

Section: The Zolotarev Metricmentioning

confidence: 99%

Refined quicksort asymptotics

Neininger

2013

Random Struct Algorithms

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The complexity of the Quicksort algorithm is usually measured by the number of key comparisons used during its execution. When operating on a list of n data, permuted uniformly at random, the appropriately normalized complexity Yn is known to converge almost surely to a non-degenerate random limit Y. This assumes a natural embedding of all Yn on one probability space, e.g., via random binary search trees. In this note a central limit theorem for the error term in the latter almost sure convergence is shown:where N denotes a standard normal random variable.

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Approximation of Distributions of Sums of Independent Random Variables with Values in Infinite-Dimensional Spaces

Cited by 73 publications

References 10 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

Profiles of Random Trees: Limit Theorems for Random Recursive Trees and Binary Search Trees

Refined quicksort asymptotics

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