Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-27T02:21:20.316Z Has data issue: false hasContentIssue false
  • English
  • Français

On the 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 Statistical Institute, Calcutta, India
Get access Rights & Permissions [Opens in a new window]

Extract

Let Zn = n−½(X1 + X2 + … + Xn), where {Xn} is a sequence of independent and identically distributed random variables with EX1 = 0, and a common distribution function F and characteristic function ω. Suppose |ω|r is integrable for some integer r ≥ 1. For all n ≥ r, then Zn has a probability density function fn obtained by using the inversion formula.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 76 , Issue 1 , July 1974 , pp. 307 - 312
Copyright
Copyright © Cambridge Philosophical Society 1974

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Feller, W.An introduction to probability theory and its applications, vol. II (John Wiley, 1966).Google Scholar
(2)Högliind, Thomas. On the convergence of convolutions of distributions with regularly varying tails. Z. Wahrscheinlichkeitstheorie und Verw. Gebeite 15 (1970), 263–272.Google Scholar
(3)Petrov, V. V.On local limit theorems for sums of independent random variables. Theor. probability appl. IX (1964), 312–320.CrossRefGoogle Scholar
(4)Smith, Walter L. and Basu, Sujit K.General moment functions and a density version of the Central Limit Theorem. Institute of Statistics (University of North Carolina) Mimeo Series no. 847. Proc. Cambridge Philos. Soc. 75 (1974), 365.CrossRefGoogle Scholar