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A local limit theorem for attractions under a stable law

Published online by Cambridge University Press:  24 October 2008

Sujit K. Basu  and
Makoto Maejima
Sujit K. Basu
Affiliation:
Indian Institute of Management, Calcutta, and Keio University, Tokyo
Makoto Maejima
Affiliation:
Indian Institute of Management, Calcutta, and Keio University, Tokyo
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Abstract

Let {Xn} be a sequence of independent random variables each having a common d.f. V1. Suppose that V1 belongs to the domain of normal attraction of a stable d.f. V0 of index α 0 ≤ α ≤ 2. Here we prove that, if the c.f. of X1 is absolutely integrable in rth power for some integer r > 1, then for all large n the d.f. of the normalized sum Zn of X1, X2, …, Xn is absolutely continuous with a p.d.f. vn such that

as n → ∞, where v0 is the p.d.f. of Vo.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 87 , Issue 1 , January 1980 , pp. 179 - 187
Copyright
Copyright © Cambridge Philosophical Society 1980

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References

REFERENCES

(1)Basu, Sujit K.On a local limit theorem concerning variables in the domain of normal attraction of a stable law of index α, 1 ≤ α ≤ 2. Ann. Probability 4 (1976), 486–489.CrossRefGoogle Scholar
(2)Basu, Sujit K.A non-uniform rate of convergence in a local limit theorem. Math. Proc. Cambridge Philos. Soc. 84 (1978), 351–360.CrossRefGoogle Scholar
(3)Feller, W.On regular variation and local limit theorems. Proc. Fifth Berkeley Symp. Math. Statist. Prob. 2 (1967), 373–388.Google Scholar
(4)Gnedenko, B. V. and Kolmogorov, A. N.Limit distributions of sums of independent random variables (revised ed.). (Reading, Mass., Addison-Wesley, 1968.)Google Scholar
(5)Ibragimov, I. A. and Linnik, Yu VIndependent and stationary sequences of random variables. (Groningen, Wolters-Noordhoff Publishing, 1971.)Google Scholar
(6)Smith, W. L. and Basu, Sujit K.General moment functions and a density version of the Central Limit Theorem. Proc. Cambridge Philos. Soc. 75 (1974), 365–381.CrossRefGoogle Scholar