The Annals of Probability

On Approximating Probabilities for Small and Large Deviations in $\mathbb{R}^d$

J. Robinson, T. Hoglund, L. Holst, and M. P. Quine

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A unified approach to approximations of probabilities for sums of $n$ independent random vectors in $\mathbb{R}^d$ is presented based on the Edgeworth expansion of exponentially shifted vectors together with explicit bounds on the errors. Weak conditions are given under which the error bounds may be written as simple order terms in $n$. These results are used in particular to examine approximations to conditional probabilities giving a general method of approximation for these. A number of important special cases are discussed and examined numerically.

Article information

Ann. Probab., Volume 18, Number 2 (1990), 727-753.

First available in Project Euclid: 19 April 2007

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Zentralblatt MATH identifier


Primary: 60F05: Central limit and other weak theorems
Secondary: 60F10: Large deviations 62E20: Asymptotic distribution theory

Edgeworth expansions saddlepoint approximations large deviations limit theorems conditional probabilities


Robinson, J.; Hoglund, T.; Holst, L.; Quine, M. P. On Approximating Probabilities for Small and Large Deviations in $\mathbb{R}^d$. Ann. Probab. 18 (1990), no. 2, 727--753. doi:10.1214/aop/1176990856.

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