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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
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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

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