The Annals of Probability

A Note on a Limit Theorem for Differentiable Mappings

K. A. Borovkov

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The main purpose of this paper is to draw attention to a simple and useful general "continuity theorem" type result, from which a great deal of limit theorems follow as almost immediate consequences. As an example, we give a new very short and transparent proof of the recent result by H. Teicher and C. Hagwood (A multidimensional CLT for maxima of normed sums); in fact, a much more general assertion is proved here. Another application of the main result establishes a correspondence between the convergence of empirical and quantile processes. (A similar result holds for the renewal and partial sums processes.)

Article information

Ann. Probab., Volume 13, Number 3 (1985), 1018-1021.

First available in Project Euclid: 19 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60F05: Central limit and other weak theorems

Weak convergence and mappings weak invariance principle


Borovkov, K. A. A Note on a Limit Theorem for Differentiable Mappings. Ann. Probab. 13 (1985), no. 3, 1018--1021. doi:10.1214/aop/1176992924.

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