## Annals of Probability

- Ann. Probab.
- Volume 3, Number 6 (1975), 992-999.

### An Approximation Theorem for Convolutions of Probability Measures

#### Abstract

An extension of the usual problem of bounding the total variation of the difference of two probability measures is considered for certain convolutions of probability measures on a measurable Abelian group. The result is a fairly general approximation theorem which also yields an $L_p$ approximation theorem and a large deviation result in some special cases. A limit theorem in equally general setting is proved as a consequence of the main theorem. As the convolutions of probability measures under consideration reduce to the Poisson binomial distribution as a special case, an alternative proof of the approximation theorem in this special case is discussed.

#### Article information

**Source**

Ann. Probab., Volume 3, Number 6 (1975), 992-999.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176996224

**Digital Object Identifier**

doi:10.1214/aop/1176996224

**Mathematical Reviews number (MathSciNet)**

MR383483

**Zentralblatt MATH identifier**

0358.60010

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60B10: Convergence of probability measures

Secondary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 60F05: Central limit and other weak theorems 60F10: Large deviations

**Keywords**

Approximation theorem convolutions probability measures $L_p$ approximation large deviation Poisson binomial distribution Poisson approximation

#### Citation

Chen, Louis H. Y. An Approximation Theorem for Convolutions of Probability Measures. Ann. Probab. 3 (1975), no. 6, 992--999. doi:10.1214/aop/1176996224. https://projecteuclid.org/euclid.aop/1176996224