Metrika

, Volume 31, Issue 1, pp 245–252 | Cite as

A local limit theorem for attraction to the standard normal law: The case of infinite variance

Short title: A local limit theorem
  • S. K. Basu
Publication
  • 49 Downloads

Summary

Let ≏Xn≃ be a sequence of independent random variables each having a common dfF. Suppose thatF belongs to the domain of attraction of the standard normal law. Here we show that if the cf ofX1 is absolutely integrable in ther-th power for some integerr≧1, then for all largen, the df of the normalized sumZn ofX1,X2, ...,Xn is absolutely continuous with a pdffn such that asn→∞,
$$\mathop {\sup }\limits_{ - \infty< x< \infty } (1 + |x|)^\beta |f_n (x) - \phi (x)| = o(1)$$
for every β<2 or β≦2 according asEX 1 2 =∞ orEX 1 2 <∞, π being the standard normal pdf.

Keywords

Central Limit Theorem Independent Random Variable Cambridge Philos Infinite Variance Local Limit Theorem 

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References

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Copyright information

© Physica-Verlag 1984

Authors and Affiliations

  • S. K. Basu
    • 1
  1. 1.Dept. of StatisticsUniversity of North Carolina at Chapel HillChapel HillUSA

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