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On an inequality of Heyde

  • R. J. Tomkins (a1)


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 X 1, X 2, ··· 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 c 1, c 2, ··· be a non-increasing sequence of positive numbers.



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[1] Gnedenko, B. V. (1963) The Theory of Probability. Chelsea, New York.
[2] Hájek, J. and Rényi, A. (1955) Generalization of an inequality of Kolmogorov. Acta Math. Acad. Sci. Hung. 6, 281283.
[3] Heyde, C. C. (1968) An extension of the Hájek-Rényi inequality for the case without moment conditions. J. Appl. Prob. 5, 481483.

On an inequality of Heyde

  • R. J. Tomkins (a1)


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