Summary
Let x 1,..., x n be a sequence of independent random variables with the common distribution F. Suppose E x k=0 and that F belongs to the domain of attraction of the normal distribution. Under conditions which do not involve the existence of any particular moment we show that
uniformly in x, provided the norming constants a 1, a 2,... are properly chosen. Here Φ is the standard normal distribution and Ω a certain operator (depending on F).
The local counterparts are also treated.
Article PDF
Similar content being viewed by others
References
Bergström, H.: On distribution functions with a limiting stable distribution function. Ark. Mat. 2, 463–474 (1953).
Esséen, C.G.: Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian law. Acta math. 77, 1–125 (1945).
—: On the remainder term in the central limit theorem. Ark. Mat. 8, 7–15 (1969).
Feller, W.: An introduction to probability theory and its applications, Vol. II. New York: Wiley 1966.
—: On the Berry-Esséen theorem. Z. Wahrscheinlichkeitstheorie verw. Geb. 10, 261–268 (1968).
Heyde, C.C.: A contribution to the theory of large deviations for sums of independent random variables. Z. Wahrscheinlichkeitstheorie verw. Geb. 7, 303–308 (1967).
—: On the influence of moments on the rate of convergence to the normal distribution. Z. Wahrscheinlichkeitstheorie verw. Geb. 8, 12–18 (1967).
—: On large deviation problems for sums of random variables which are not attracted to the normal law. Ann. math. Statistics 38, 1575–1578 (1967).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Höglund, T. On the convergence of convolutions of distributions with regularly varying tails. Z. Wahrscheinlichkeitstheorie verw Gebiete 15, 263–272 (1970). https://doi.org/10.1007/BF00533297
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00533297