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Inequalities for the tail probabilities of weighted sums of independent random variables with applications to rates of convergence to zero

Published online by Cambridge University Press:  14 November 2011

J. E. A. Dunnage
J. E. A. Dunnage
Affiliation:
Chelsea College (University of London), London SW3 6LX
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Synopsis

We obtain inequalities for where Wn = anlX1 + … + annXn, the Xr being independent random variables and the Mn being certain truncated means. We then use these inequalities to study the rate at which this probability tends to zero as N→ ∞, noting that in the special case Wn = (X1 + … + Xn)/n, we obtain the estimate given by L. E. Baum and M. Katz which they show is, in a sense, best possible.

A desire to find an inequality which would lead to the result of Baum and Katz was, indeed, the impetus behind this paper.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 93 , Issue 1-2 , 1982 , pp. 111 - 121
Copyright
Copyright © Royal Society of Edinburgh 1982

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References

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