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Tail sums of convergent series of independent random variables

Published online by Cambridge University Press:  24 October 2008

Andrew D. Barbour
Andrew D. Barbour
Affiliation:
Statistical Laboratory, 16 Mill Lane, Cambridge, Great Britain
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Extract

Let X1, X2, … be a sequence of independent random variables such that, for each n ≥ 1, EXn = 0 and and assume that then converges almost surely as N → ∞. Let and let Fn(x) denote the distribution function of Xn. Loynes (2) observed that the sequence {Sn} is a reversed martingale, and applied his central limit theorem to it: however, stronger results are obtainable, in precise duality with the classical theory of partial sums of independent random variables. These results describe the fluctuations of the sequence {Sn}, and hence the way in which converges to its limit.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 75 , Issue 3 , May 1974 , pp. 361 - 364
Copyright
Copyright © Cambridge Philosophical Society 1974

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References

REFERENCES

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