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On an inequality of Heyde

Published online by Cambridge University Press:  14 July 2016

R. J. Tomkins
R. J. Tomkins*
Affiliation:
University of Saskatchewan, Regina Campus
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Extract

In [3] Heyde proposed an extended version of the well-known Hájek-Rényi inequality to include random variables without finite moments. The purpose of this note is to point out that the theorem in [3] is in error, and to prove the following theorem in its stead:

Theorem. Let X1, X2, ··· be any sequence of random variables, and define be independent random variables, each with mean zero and finite variance, and define Zk = Xk - Yk for k ≧ 1. Let c1, c2, ··· be a non-increasing sequence of positive numbers.

Type
Short Communications
Information
Journal of Applied Probability , Volume 8 , Issue 2 , June 1971 , pp. 428 - 429
Copyright
Copyright © Applied Probability Trust 1970 

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References

[1] Gnedenko, B. V. (1963) The Theory of Probability. Chelsea, New York.Google Scholar
[2] Hájek, J. and Rényi, A. (1955) Generalization of an inequality of Kolmogorov. Acta Math. Acad. Sci. Hung. 6, 281–283.Google Scholar
[3] Heyde, C. C. (1968) An extension of the Hájek-Rényi inequality for the case without moment conditions. J. Appl. Prob. 5, 481–483.CrossRefGoogle Scholar