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A theorem on the cumulative product of independent random variables

Published online by Cambridge University Press:  24 October 2008

Harold Ruben
Harold Ruben
Affiliation:
Manchester UniversityEngland Columbia UniversityNew York
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Extract

1. Introductory discussion and summary. Consider a sequence {ui} of independent real or complex-valued random variables such that E(ui) = 1, and a sequence of mutually exclusive events S1, S2,…, such that Si depends only on u1, u2, …,ui, with ΣP(Sj) = 1. Define the random variable n = n(u1, u2,…) = m when Sm occurs. We shall obtain the necessary and sufficient conditions under which

referred to as the product theorem.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 55 , Issue 4 , October 1959 , pp. 333 - 337
Copyright
Copyright © Cambridge Philosophical Society 1959

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References

REFERENCES

(1)Barnard, G. A.On the Fisher-Behrens test. Biometrika, 37 (1950), 203–7.CrossRefGoogle ScholarPubMed
(2)Bahadur, R. R.A note on the fundamental identity of sequential analysis. Ann. Math. Stat. 29 (1958), 534–43.CrossRefGoogle Scholar
(3)Bellman, R.On a generalization of the fundamental identity of Wald. Proc. Camb. Phil. Soc. 53 (1957), 257–9.Google Scholar
(4)Blackwell, D.On an equation of Wald. Ann. Math. Stat. 17 (1946), 84–7.CrossRefGoogle Scholar
(5)Blackwell, D.Extension of a renewal theorem. Pacific J. Math. 3 (1953), 315–20.Google Scholar
(6)Blackwell, D. and Girshick, M. A.On functions of sequences of independent chance vectors with applications to the problem of the ‘random walk’ in k dimensions. Ann. Math. Stat. 17 (1946), 310–17.CrossRefGoogle Scholar
(7)Blom, G.A generalization of Wald's Fundamental Identity. Ann. Math. Stat. 20 (1949), 439–44.CrossRefGoogle Scholar
(8)Doob, J. L.Stochastic processes (New York, 1953).Google Scholar
(9)Wald, A.On cumulative sums of random variables. Ann. Math. Stat. 15 (1944), 283–96.CrossRefGoogle Scholar
(10)Wald, A.Sequential tests of statistical hypotheses. Ann. Math. Stat. 15 (1945), 117–86.Google Scholar
(11)Wolfowitz, J.The efficiency of sequential estimates and Wald's equation for sequential procedures. Ann. Math. Stat. 18 (1947), 215–30.Google Scholar