Vol. 49, No. 1, 1973

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ISSN: 0030-8730
On the renewal function when some of the mean renewal lifetimes are infinite

Makoto Maejima

Vol. 49 (1973), No. 1, 177–184
Abstract

Let {Xi,i = 1,2,} be a sequence of independent and nonnegative random variables with the distribution function Fi(x). Some of 0xdFi(x) may be infinite. Let H(t) be the renewal function. The main object of this note is to show that in order to have the asymptotic relation H(t)∕t 1∕L(t) as t →∞, it is necessary and sufficient that μ(t) L(t) as t →∞, where L(t) is a function of slow growth and μ(t) = lim2 →∞(1∕n) i=1nμi(t)i(t) being 0t[1 Fi(x)]dx, is supposed to exist uniformly in t.

Mathematical Subject Classification 2000
Primary: 60K05
Milestones
Received: 3 March 1973
Published: 1 November 1973
Authors
Makoto Maejima