Abstract
This paper investigates the defective renewal equations under the nonconvolution equivalent distribution class. The asymptotics of the solution to the defective renewal equations have been given for the heavy-tailed and light-tailed cases, respectively.
Highlights
This paper will consider the defective renewal equation xZ (x) = z (x) + q ∫ Z (x − y) F, x ≥ 0, (1)where F is a proper distribution on [0, ∞), z(x) ≥ 0 is a known and locally bounded function on [0, ∞), and 0 < q < 1
That F may not belong to the convolution equivalent distribution class, this paper obtains the asymptotics of the solution Z(x)
Say that the distribution V belongs to the class L(γ) for some γ ≥ 0, if for any t ∈ (−∞, ∞), V (x − t) ∼ eγtV (x), (5)
Summary
5.2 of Cui et al [9], they obtained the asymptotics of Z(x) under the condition that F ∈ S and F ∈ S(γ) for some γ > 0 with F(γ) = ∫−∞∞ eγuF(du) < 1, respectively. Beyond the convolution equivalent distribution classes, there exist some other distributions. How to estimate the asymptotics of the solution Z(x) for the nonconvolution equivalent distribution F will be an interesting question. That F may not belong to the convolution equivalent distribution class, this paper obtains the asymptotics of the solution Z(x).
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