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Saddle point approximation for the distribution of the sum of independent random variables

Published online by Cambridge University Press:  01 July 2016

Robert Lugannani  and
Stephen Rice
Robert Lugannani*
Affiliation:
University of California, San Diego
Stephen Rice*
Affiliation:
University of California, San Diego
*
∗Postal address: Department of Applied Physics and Information Science, University of California, San Diego, La Jolla, CA 92093, U.S.A.
∗Postal address: Department of Applied Physics and Information Science, University of California, San Diego, La Jolla, CA 92093, U.S.A.
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Abstract

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In the present paper a uniform asymptotic series is derived for the probability distribution of the sum of a large number of independent random variables. In contrast to the usual Edgeworth-type series, the uniform series gives good accuracy throughout its entire domain. Our derivation uses the fact that the major components of the distribution are determined by a saddle point and a singularity at the origin. The analogous series for the probability density, due to Daniels, depends only on the saddle point. Two illustrative examples are presented that show excellent agreement with the exact distributions.

Keywords

SADDLE POINT APPROXIMATIONSUM OF INDEPENDENT RANDOM VARIABLESUNIFORM ASYMPTOTIC SERIES
Type
Research Article
Information
Advances in Applied Probability , Volume 12 , Issue 2 , June 1980 , pp. 475 - 490
Copyright
Copyright © Applied Probability Trust 1980 

Footnotes

This research was sponsored by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant No. AFOSR 74-2689.

References

1. Abramowitz, M. and Stegun, I. A. (1972) Handbook of Mathematical Functions. Dover, New York.Google Scholar
2. Bleistein, N. (1966) Uniform asymptotic expansions of integrals with stationary point near algebraic singularity. Comm. Pure Appl. Math. 19, 353–370.CrossRefGoogle Scholar
3. Bleistein, N. and Handelsman, R. A. (1975) Asymptotic Expansion of Integrals. Holt, Rinehart and Winston, New York.Google Scholar
4. Chernoff, H. (1952) A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Statist. 23, 493–507.CrossRefGoogle Scholar
5. Daniels, H. E. (1954) Saddlepoint approximations in statistics. Ann. Math. Statist. 25, 631–650.CrossRefGoogle Scholar
6. Olver, F. W. J. (1974) Asymptotics and Special Functions. Academic Press, New York.Google Scholar
7. Petrov, V. V. (1973) Sums of Independent Random Variables, trans. Brown, A. A. Springer-Verlag, New York.Google Scholar
8. Rice, S. O. (1968) Uniform asymptotic expansions for saddle point integrals—Applications to a probability distribution occurring in noise theory. Bell Syst. Tech. J. 47, 1971–2013.CrossRefGoogle Scholar
9. Roberts, J. B. (1972) Distribution of the response of linear systems to Poisson distributed random impulses. J. Sound and Vibration 24, 23–34.CrossRefGoogle Scholar
10. Saulis, L. (1969) Asymptotic expansions of probabilities of large deviations (in Russian). Litovsk. Mat. Sb. 9, 605–625.Google Scholar
11. Van Der Waerden, B. L. (1951) On the method of saddle points. Appl. Sci. Res. B2, 33–45.Google Scholar