Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-23T13:28:14.474Z Has data issue: false hasContentIssue false
  • English
  • Français

Two probability theorems and their application to some first passage problems

Published online by Cambridge University Press:  09 April 2009

C. C. Heyde
C. C. Heyde
Affiliation:
The Australian National University, Canberra
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let Xi, i = 1, 2, 3,··· be a sequence of independent and identically distributed random variables and write Sn = X1+X2+…+Xn. If the mean of Xi is finite and positive, we have Pr(Sn ≦ x) → 0 as n → ∞ for all x1 – ∞ < x < ∞ using the weak law of large numbers. It is our purpose in this paper to study the rate of convergence of Pr(Sn ≦ x) to zero. Necessary and sufficient conditions are established for the convergence of the two series where k is a non-negative integer, and where r > 0. These conditions are applied to some first passage problems for sums of random variables. The former is also used in correcting a queueing Theorem of Finch [4].

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 4 , Issue 2 , May 1964 , pp. 214 - 222
Copyright
Copyright © Australian Mathematical Society 1964

References

[1]Baum, L. E., Katz, M. L., and Read, R. R., Exponential convergence rates for the law of large numbers, Trans. Amer. Math. Soc., 102 (1962), 187–199.CrossRefGoogle Scholar
[2]Derman, C., and Robbins, H., The strong law of large numbers when the first moment does not exist, Proc. Nat. Acad. Sci. U.S.A., 41 (1955), 586–587.CrossRefGoogle Scholar
[3]Erdös, P., On a theorem of Hsu and Robbins, Ann. Math. Statist., 20 (1949), 286–291.CrossRefGoogle Scholar
[4]Finch, P. D., On the busy period in the queueing system GI/G/1, J. Austral. Math. Soc., 2 (1961), 217–228.CrossRefGoogle Scholar
[5]Katz, M. L., The probability in the tail of a distribution, Ann. Math. Statist., 34 (1963), 312–318.Google Scholar
[6]Kemperman, J. H. B., The passage problem for a stationary Markov chain, Univ. of Chicago Press (1961).CrossRefGoogle Scholar
[7]Lukacs, E., Characteristic functions, Griffin, London (1960).Google Scholar
[8]Spitzer, F., A combinatorial lemma and its applications to probability theory, Trans. Amer. Math. Soc., 82 (1956), 323–339.CrossRefGoogle Scholar
[9]Stein, C., A note on cumulative sums, Ann. Math. Statist., 17 (1946), 498–499.CrossRefGoogle Scholar