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On moment conditions for supremum of normed sums of martingale differences

Published online by Cambridge University Press:  17 April 2009

Wi Chong Ahn ,
Bong Dae Choi  and
Soo Hak Sung
Wi Chong Ahn
Affiliation:
Department of Mathematics Education, Koomin University Seoul, 136-702, Korea
Bong Dae Choi
Affiliation:
Department of Applied Mathematics, Korea Advanced Institute of Science and Technology, Seoul 130-650, Korea
Soo Hak Sung
Affiliation:
Coding Technology Section I, Electronics and Telecommunications Research Institute, Taejeon 305-606, Korea
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Abstract

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Let {Sn, n ≥ 1} denote the partial sum of sequence (Xn) of identically distributed martingale differences. It is shown that E|X1|q (lg |X1|)r < ∞ implies E(sup((lg n)pr/q/npr/q)|Sn|p) < ∞, where 1 < p < 2, p < q, r ∈ R and lg x = max{1, log+x} For the independent identically distributed case, the converse of the above statement holds.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 43 , Issue 2 , April 1991 , pp. 273 - 277
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Choi, B. D. and Sung, S. H., ‘On moment conditions for supremum of normed sums’, Stochastic Process. Appl. 26 (1987), 99–106.CrossRefGoogle Scholar
[2]Stout, W. F., Almost Sure Convergence (Academic Press, New York, 1974).Google Scholar