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Some results using general moment functions

Published online by Cambridge University Press:  09 April 2009

Walter L. Smith
Walter L. Smith
Affiliation:
University of North CarolinaChapel Hill, N.C.
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Let {Xn} be a sequence fo independent and identically distributed random variables such that 0 <μ = εXn ≦ + ∞ and write Sn = X1+X2+ … +Xn. Letv ≧ 0 be an integer and let M(x) be a non-decreasing function of x ≧ 0 such that M(x)/x is non-increasing and M(0) > 0. Then if ε|X1νM(|X1|) < ∞ and μ < ∞ it follows that ε|Sn|νM(|Sn|) ~ (nμ)vM(nμ) as n → ∞. If μ = ∞ (ν = 0) then εM(|Sn|) = 0(n). A variety of results stem from this main theorem (Theorem 2), concerning a closure property of probability generating functions and a random walk result (Theorem 1) connected with queues.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 10 , Issue 3-4 , November 1969 , pp. 429 - 441
Copyright
Copyright © Australian Mathematical Society 1969

References

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