Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-26T07:28:49.648Z Has data issue: false hasContentIssue false
  • English
  • Français

Extensions of the Hájek–Rényi inequality to moments of higher order

Published online by Cambridge University Press:  24 October 2008

J. E. A. Dunnage
J. E. A. Dunnage
Affiliation:
Chelsea College of Science and Technology, London SW3 6LX
Get access Rights & Permissions [Opens in a new window]

Extract

1. Introduction. Bernoulli trials. Consider a sequence of Bernoulli trials. Let p, assumed to satisfy 0<p < 1, be the probability of success at any given trial and let q = 1–p. If Nn is the number of successes in the first n trials, it is well known that Nn/n→p almost surely as n→∞ so that for every ∈> 0,

as n→∞, and it is clearly of great interest to know quantitatively how this probability depends upon n and ∈.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 72 , Issue 1 , July 1972 , pp. 67 - 75
Copyright
Copyright © Cambridge Philosophical Society 1972

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)Bernstein, S. N.Probability Theory, 4th ed. (Goztehizdat, Moscow, 1946).Google Scholar
(2)Bickel, P. J. A.Hájek–Rényi extension of Lévy's inequality and some applications. Acta Math. Acad. Sci. Hungar. 21 (1970), 199–206.Google Scholar
(3)Dunnage, J. E. A.Khintchine's lemma on sums of random variables. J. London Math. Soc. (2) 2 (1970), 563–566.CrossRefGoogle Scholar
(4)Hájek, J. and Rényi, A.Generalization of an inequality of Kolmogorov. Acta Math. Acad. Sci. Hungar. 6 (1955), 281–283.CrossRefGoogle Scholar
(5)Loète, M.Probability Theory, 3rd ed. (Van Nostrand, New York, 1963).Google Scholar
(6)Rényi, A.Probability Theory. (North-Holland Pub. Co., Amsterdam, 1970.)Google Scholar