Abstract
Consider a continuous time process {Yt=ZNt, t≥0}, where {Zn} is a supercritical Galton–Watson process and {Nt} is a Poisson process which is independent of {Zn}. Let τn be the n-th jumping time of {Yt}, we obtain that the typical rate of growth for {τn} is n/λ, where λ is the intensity of {Nt}. Probabilities of deviations n-1τn-λ-1>δ are estimated for three types of positive δ.
Highlights
Obtain that the typical are estimated for three rate of growth for {τn} types of positive δ
In a recent manuscript [3] the authors there consider the asymptotic properties of log Yt
Let {pk, k ≥ 0} be the offspring distribution of the branching process with mean m = ∑k kpk ∈ (1, ∞); we distinguish between the Shroder case and the Bottcher case depending on whether p0 + p1 > 0 or p0 + p1 = 0
Summary
We deal with the asymptotic theory for the jumping times of PRIBP defined as follows. {τn} satisfies the law of large number and the central limit theorem; that is, τn/n a→.e. λ−1 and λ√n(τn/n − λ−1) →d N(0, 1) when n → ∞, where N(0, 1) is standard normal distribution. By Theorem 2, the rate function of τn/n coincides with that of Tn/n for x ≤ (λp1)−1, but differences appeared for large x; see Figure 2 for example. As in the case of large deviation principle, based on the Gartner-Ellis theorem (see [7], page 44), we have the following moderate deviation principle. Basic facts on Gartner-Ellis theorem are given in the Appendix
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