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

Short title: A local limit theorem

Publication

First Online:

Received:

Revised:

- 50 Downloads

## Summary

Let ≏ for every β<2 or β≦2 according as

*X*_{n}≃ be a sequence of independent random variables each having a common df*F*. Suppose that*F*belongs to the domain of attraction of the standard normal law. Here we show that if the cf of*X*_{1}is absolutely integrable in the*r*-th power for some integer*r*≧1, then for all large*n*, the df of the normalized sum*Z*_{n}of*X*_{1},*X*_{2}, ...,*X*_{n}is absolutely continuous with a pdf*f*_{n}such that as*n*→∞,$$\mathop {\sup }\limits_{ - \infty< x< \infty } (1 + |x|)^\beta |f_n (x) - \phi (x)| = o(1)$$

*EX*_{1}^{2}=∞ or*EX*_{1}^{2}<∞, π being the standard normal pdf.## Keywords

Central Limit Theorem Independent Random Variable Cambridge Philos Infinite Variance Local Limit Theorem
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## References

*Basu, S.K.*: A non-uniform rate of convergence in a local limit theorem. Math. Proc. Cambridge Philos. Soc.**84**, 1978, 351–359.MathSciNetCrossRefMATHGoogle Scholar*Basu, S.K.*, and*M. Maejima*: A local limit theorem for attraction under a stable law. Math. Proc. Cambridge Philos. Soc.**87**, 1980, 179–187.MathSciNetCrossRefMATHGoogle Scholar*Basu, S.K., M. Maejima*, and*N. Patra*: A non-uniform rate of convergence in a local limit theorem concerning variables in the domain of normal attraction of a stable law. Yokohama Math. J.**27**, 1979, 63–72.MathSciNetMATHGoogle Scholar*Basu, S.K.*, and*N. Patra*: A generalised version of a local limit theorem for attractions under a stable law. Forthcoming.Google Scholar*Feller, W.*: An Introduction to Probability Theory and Its Applications. Vol. II. New York-London-Sydney 1966.Google Scholar*Gnedenko, B.V.*, and*A.N. Kolmogorov*: Limit Distributions of Sums of Independent Random Variables. MA-Palo Alto-London 1954.Google Scholar*Ibragimov, I.A.*, and*Yu.V. Linnik*: Independent and Stationary Sequences of Random Variables. Groningen 1971.Google Scholar*Smith, W.L.*: A frequency function form of the central limit theorem. Proc. Cambridge Philos. Soc.**49**, 1953, 462–472.MathSciNetCrossRefMATHGoogle Scholar*Smith, W.L.*, and*S.K. Basu*: Central moment functions and a density version of the central limit theorem. Proc. Cambridge Philos. Soc.**75**, 1974 365–381.MathSciNetCrossRefMATHGoogle Scholar*Titchmarsh, J.*: Theory of Fourier Integrals. Oxford 1937.Google Scholar

## Copyright information

© Physica-Verlag 1984