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On one-sided boundedness of normed partial sums

Published online by Cambridge University Press:  17 April 2009

R. A. Maller
R. A. Maller
Affiliation:
Division of Mathematics and Statistics, CSIRO, Private Bag, PO Wembley, Western Australia 6014, Australia.
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This paper gives a very general sufficient condition for the existence of constants B(n), C(n) for which either almost surely or almost surely, where Sn = X1 + X2 + … + Xn and Xi are independent and identically distributed random variables. The theorem is closely connected with results of Klass and Teicher on the one-sided boundedness of Sn, with the relative stability of Sn, and with a generalised law of the iterated logarithm due to Kesten. For non negative Xi the sufficient condition is shown to be necessary, and the results are partially generalised to the case when Xi form a stationary m-dependent sequence. Some connections with a generalised type of regular variation and with domains of partial attraction are also noted.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 21 , Issue 3 , June 1980 , pp. 373 - 391
Copyright
Copyright © Australian Mathematical Society 1980

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