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Asymptotic analysis and local lattice central limit theorems

Published online by Cambridge University Press:  14 July 2016

P.A.P. Moran
P.A.P. Moran*
Affiliation:
The Australian National University, Canberra
*
∗ Postal address: Department of Statistics, The Australian National University, P. O. Box 4, Canberra, Australia 2600.
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Abstract

Methods of evaluating the coefficients of high powers of functions defined by power series with positive coefficients are considered. Such methods, which were originally used by Laplace, can, for example, be used to obtain asymptotic formulae for Stirling numbers. They are equivalent to using local lattice central limit theorems. An alternative method using direct numerical integration on a contour integral giving the required coefficient is described. Exact bounds for the accuracy of this method can often be obtained by considerations of the unimodality of discrete distributions. The results are illustrated using convolutions of the rectangular and logarithmic distributions.

Keywords

ASYMPTOTIC ANALYSISLOCAL LATTICE CENTRAL LIMIT THEOREMSTIRLING NUMBERS
Type
Research Papers
Information
Journal of Applied Probability , Volume 16 , Issue 3 , September 1979 , pp. 541 - 553
Copyright
Copyright © Applied Probability Trust 1979 

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References

Abramowitz, M. and Stegum, I. A. (1965) Handbook of Mathematical Functions. Dover, New York.Google Scholar
Fréchet, M. (1937) Recherches Théoriques Modernes sur la Théorie des Probabilités. Premier Livre. Généralités sur les Probabilités Variables Aléatoires. Gauthier Villars, Paris.Google Scholar
Gamkrelidze, N. G. (1966) On the speed of convergence in the local limit theorem for lattice distributions. Theory Prob. Appl. 11, 114–125.Google Scholar
Good, I. J. (1961) An asymptotic formula for the differences of the powers at zero. Ann. Math. Statist. 32, 249–256.Google Scholar
Hammersley, J. M. (1951) The sums of products of the natural numbers. Proc. London Math. Soc. (3) 1, 435–452.Google Scholar
Ibragimov, I. A. (1956) On the composition of unimodal distributions. Theory Prob. Appl. 1, 255–260.Google Scholar
Moser, L. and Wyman, M. (1958a) Asymptotic development of the Stirling numbers of the first kind. J. London Math. Soc. 33, 133–146.Google Scholar
Moser, L. and Wyman, M. (1958b) Stirling numbers of the second kind. Duke Math. J. 25, 29–43.Google Scholar
Patil, G. P. and Bildikar, S. (1966) On minimum variance unbiased estimation for the logarithmic series distribution. Sankhya A 28, 239–250.Google Scholar
Patil, G. P. and Wani, J. K. (1965) On certain structural properties of the logarithmic series distribution and the first type Stirling distribution. Sankhya A 27, 271–292.Google Scholar
Patil, G. P., Kamat, A. R. and Wani, J. K. (1964) The structure and statistics of the logarithmic distribution and related tables. Aerospace Research Laboratory.Google Scholar
Petrov, V. V. (1975) Sums of Independent Random Variables, trans. Brown, A. A. Springer-Verlag, Berlin.Google Scholar
Sheynin, O. (1972–3) Finite random sums (a historical essay). Archive for the History of Exact Sciences 9, 275–305.Google Scholar
Watterson, G. A. and Guess, H. A. (1977) Is the most frequent allele the oldest? Theoret. Popn. Biol. 11, 141–160.Google Scholar
Yates, F. (1948) Systematic sampling. Phil. Trans. R. Soc. A 241, 345–377.Google Scholar