Annals of Probability

An Approximation Theorem for Convolutions of Probability Measures

Louis H. Y. Chen

Full-text: Open access


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

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

First available in Project Euclid: 19 April 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


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

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


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.

Export citation