Deviations for Martingale Convergence of a Branching Process with Random Index
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Deviations for Martingale Convergence of a Branching Process with Random Index Zhenlong Gao1
· Min Wang1 · Huili Zhang1
Received: 1 February 2019 / Revised: 5 December 2019 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract The purpose of this paper is to obtain large deviations, moderate deviations and normal deviations for the convergence of martingale generated by a supercritical Galton– Watson process with a Poisson process as its time index. Harmonic moments and Shröder index play an important role in the proofs. Keywords Large deviations · Moderate deviations · Random indexed branching process Mathematics Subject Classification 60J80 · 60F10
1 Introduction Consider a Galton–Watson process (GW) {Z n } with offspring distribution { pi } and offspring mean m := i i pi . The ratios {Wn := Z n /m n } constitute a nonnegative martingale with an almost sure limit W as n → ∞. Throughout this manuscript, we assume that Z 0 = 1, p0 = 0 and GW is supercritical, that is m > 1. Consequently, P(Z n > 0) = 1 for all n and P(W > 0) = 1. Most of the published literature about martingale {Wn } has emphasized the decay rates of the probability for deviations between Wn and W , see [5] for normal deviations and [1,13] for large deviations.
Communicated by Anton Abdulbasah Kamil. Supported by NSFC (No. 11601260).
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Zhenlong Gao [email protected] Min Wang [email protected] Huili Zhang [email protected]
1
School of Statistics, Qufu Normal University, Qufu 273165, China
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Z. Gao et al.
Let {Nt } be a Poisson process which is independent of {Z n }, the continuous time process {Yt := Z Nt } is said to be a Poisson randomly indexed branching process(PRIBP). The model of PRIBP was introduced by Epps [4] to study the evolution of stock prices. The statistical investigation on various estimates and some parameters of the process were done in [3]. Particularly, Tt := log(Yt )/(λt) is used to estimate log m, where λ is the density of underlying Poisson process. In recent manuscripts, various rates of convergence for estimator Tt have been derived, such as asymptotic normality, Berry– Esseen bound, large deviations and moderate deviations, see [6–11]. In this paper, we consider the decay rates of the probabilities of deviations in the form {l(Yt )|Rt − 1| ≥ a},
(1)
where Rt := W /W Nt and l(n) : N → R+ is a positive and nondecreasing function. Three cases of l(n) are considered in the literature. If√l(n) = O(1), then (1) is said to be a large deviation event. Secondly, if l(n) = O( n), (1) is called a normal deviation event. Finally, if √ the growth rate of l(n) is between the two above, that is, l(n) → ∞, l(n) = o( n), we say (1) belongs to a moderate deviation event. Particularly, one can chose l(n) = n 1/2−δ , 0 < δ < 1/2. Typically, harmonic moments of branching process play an important role in proving results of deviations. So we begin with the harmonic moments of {Yt }. Harmonic moments for GW were given in [13]. For a PRIBP, we distinguish between the Shröder case and the
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