Deviations for Jumping Times of a Branching Process Indexed by a Poisson Process

Consider a continuous time process {𝑌 𝑡 = 𝑍 𝑁 𝑡 , 𝑡 ≥ 0} , where {𝑍 𝑛 } is a supercritical Galton–Watson process and {𝑁 𝑡 } is a Poisson process which is independent of {𝑍 𝑛 } . Let 𝜏 𝑛 be the 𝑛− th jumping time of {𝑌 𝑡 }


Statements of the Main Results
The model of Poisson randomly indexed branching process (PRIBP) { = , ≥ 0} was introduced by [1] to study the evolution of stock prices and its statistical investigation was done in [2].
In a recent manuscript [3] the authors there consider the asymptotic properties of log . Let { , ≥ 0} be the offspring distribution of the branching process with mean = ∑ ∈ (1, ∞); we distinguish between the Shröder case and the Böttcher case depending on whether 0 + 1 > 0 or 0 + 1 = 0. In Böttcher case, it was proved in [3] that log have similar asymptotic results to the Poisson process { }. But differences appeared in Shröder case; see [4]. For subcritical and critical PRIBP, one can see [5,6] for details.
In this paper, we deal with the asymptotic theory for the jumping times of PRIBP defined as follows. For any , define where inf 0 = ∞; then Note that { } is independent of { }; one has Although ≥ for all , the growth rate of is not too fast as that of . In fact, the typical growth rate of is / by the law of large numbers and we can show that the typical growth rate of is and see the proof of Theorem 1. Thus, for almost all the path of Shröder case PRIBP, / has a limit −1 when → ∞. In the rest of this paper, we always assume that our branching process belongs to the Shröder case, 0 = 0 and 0 = 1.
We are interested in the decay rates about the probabilities of for some positive . Typically, there are three classes of to be chosen. The first one is that = √ for some fixed > 0. In this case, the event in (5) is said to be a normal deviation event. The decay rate of its probability can be characterized by the following central limit theorem. Next, if = for some fixed > 0, the event in (5) is said to be a large deviation event whose probability has an exponential convergence rate by the following large deviation principle.

Theorem 2 (LDP). For any measurable subset of
where denotes the interior of B, its closure, and +∞, ≤ 0.

Remark. By Cramér's theorem (see Theorem 2.2.3 of [7]),
/ satisfies the large deviation principle with rate function By Theorem 2, the rate function of / coincides with that of / for ≤ ( 1 ) −1 , but differences appeared for large ; see As in the case of large deviation principle, based on the Gärtner-Ellis theorem (see [7], page 44), we have the following moderate deviation principle.

Theorem 3 (MDP). For any measurable subset of ,
The rest of the paper is organized as follows. In Section 2, we prove the law of the large number and the central limit Mathematical Problems in Engineering 3 theorem. Section 3 is devoted to the proof of the large deviation principle. In Section 4, the moderate deviation principle is obtained. Basic facts on Gärtner-Ellis theorem are given in the Appendix. Proof. For any nonnegative real numbers 0 < 1 < ⋅ ⋅ ⋅ <

The Law of Large Number and the Cental Limit Theorem
For any nonnegative integers Since Poisson process { } is independent of the Galton-Watson process { }, Note that the Galton-Watson process is a Markov chain with − step transition probabilities ( , ), and summing 0 , . . . , −1 , one has Similarly, which means that PRIBP is a homogenous continuous time Markov chain.
Next, note that has a Poisson distribution with parameter > 0; we have → , see page 2 of [9]. Note that −1 ≥ ; one has which implies that / . .
, F −1 be the − field generated by 1 , . . . , −1 ; by Hölder's inequality, one has where is the indicator function of . Note that the conditional distribution of relative to 1 , . . . , −1 equals exponential distribution with parameter 4

Moderate Deviation Principle
In this section, we deal with the proof of Theorem 3. Define Lemma 7. For each ∈ , one has, Particularly, 0 ∈ Δ fl { : Δ( ) < ∞}. In addition, let Δ * be the Fenchel-Legendre transform of Δ; then and the set of exposed points of Δ * is F = .
Proof of Theorem 2. Note that the set of exposed points of Δ * is ; Theorem 3 follows from Lemma 7 and the Gärtner-Ellis theorem.