Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-30T20:09:11.110Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Chung, K. L.A Course in probability theory (Harcourt, Brace and World; New York, 1968).Google Scholar
(2)Loynes, R. M.An invariance principle for reversed martingales. Proc. Amer. Math. Soc. 25 (1970), 56–64.CrossRefGoogle Scholar
(3)Muller, D. W.Verteilungs-Invarianzprinzipien fur das starke Gesetz der groβen Zahl. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 10 (1968), 173–192.CrossRefGoogle Scholar
(4)Strassen, V.An invariance principle for the law of the iterated logarithm. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 3 (1964), 211–226.CrossRefGoogle Scholar
(5)Whitt, W.Stochastic Abelian and Tauberian theorems. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 22 (1972), 251–267.CrossRefGoogle Scholar