On the renewal function when some of the mean renewal lifetimes are infinite

Makoto Maejima

Vol. 49 (1973), No. 1, 177–184

DOI: 10.2140/pjm.1973.49.177

Abstract

Let {X_i,i = 1,2,⋯} be a sequence of independent and nonnegative random variables with the distribution function F_i(x). Some of ∫ ₀^∞xdF_i(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=1ⁿμ_i(t),μ_i(t) being ∫ ₀^t[1 −F_i(x)]dx, is supposed to exist uniformly in t.

Mathematical Subject Classification 2000

Primary: 60K05

Milestones

Received: 3 March 1973

Published: 1 November 1973