## The Annals of Probability

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

### A Note on a Limit Theorem for Differentiable Mappings

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176992924

**Digital Object Identifier**

doi:10.1214/aop/1176992924

**Mathematical Reviews number (MathSciNet)**

MR799438

**Zentralblatt MATH identifier**

0574.60047

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F17: Functional limit theorems; invariance principles

Secondary: 60F05: Central limit and other weak theorems

**Keywords**

Weak convergence and mappings weak invariance principle

#### Citation

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. https://projecteuclid.org/euclid.aop/1176992924