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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

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