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A non-uniform rate of convergence in a local limit theorem

Published online by Cambridge University Press:  24 October 2008

Sujit K. Basu
Sujit K. Basu
Affiliation:
Indian Institute of Management, Calcutta, India
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Abstract

Let {Xn} be a sequence of iid random variables. If the common charac-teristic function is absolutely integrable in mth power for some integer m ≥ 1, then Zn = n−½(X1 + … + Xn) has a pdf fn for all n ≥ m. Here we give a necessary and sufficient condition for sup as n. → ∞, where φ (x) is the standard normal pdf and M(x) is a non-decreasing function of x ≥ 0 such that M(0) > 0 and M(x)/xδ is non-increasing for 0 < δ ≤ 1.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 84 , Issue 2 , September 1978 , pp. 351 - 359
Copyright
Copyright © Cambridge Philosophical Society 1978

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References

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