Large Deviation Rates for Supercritical Branching Processes with Immigration
- PDF / 241,604 Bytes
- 11 Pages / 439.37 x 666.142 pts Page_size
- 85 Downloads / 190 Views
Large Deviation Rates for Supercritical Branching Processes with Immigration Liuyan Li1 · Junping Li1 Received: 20 November 2018 / Revised: 25 October 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract Let {X n }∞ 0 be a supercritical branching process with immigration with offspring distri∞ this paper, we assume bution { p j }∞ 0 and immigration distribution {h i }0 . Throughout ∞ j p < ∞, and h 0 < 1, that p0 = 0, p j = 1 for any j ≥ 1 , 1 < m = j j=0 ∞ m n+1 −1 −n 0 < a = j=0 j h j < ∞. We first show that Yn = m (X n − m−1 a) is a martingale and converges to a random variable Y . Secondly, we study the rates of convergence to 0 as n → ∞ of X n+1 − m > εY ≥ α P(|Yn − Y | > ε), P Xn ∞ for ε > 0 and α > 0 under various moment conditions on { p j }∞ 0 and {h i }0 . It is shown that the rates are always supergeometric under a finite moment generating function hypothesis.
Keywords Large deviation · Supercritical branching process · Immigration Mathematics Subject Classification (2010) Primary 60J27 · Secondary 60J35
1 Introduction and the Main Result Branching process is one of the most important classes of stochastic processes. Standard references are, among many others, Harris [1], Roitershtein [2] and Athreya &
B
Junping Li [email protected] Liuyan Li [email protected]
1
School of Mathematics and Statistics, Central South University, Changsha 410083, Hunan, People’s Republic of China
123
Journal of Theoretical Probability
Ney [3]. Let {X n }∞ 0 be a branching process with immigration. By Seneta [4], X n can be expressed as X 0 = 1, X n = Z n + Un(1) + Un(2) + · · · + Un(n) , n ≥ 1, (i)
where Z n is the number of direct descendants of the initial individual and Un , i = 1, 2, . . . , n, is the number of direct descendants of the individuals from the immigration at time i. The individuals of X n are independent. Thus, {Z n }∞ 0 is called a (n) classical Galton–Watson process initiated by only one ancestor, and Un (n ≥ 1) is a sequence of nonnegative integer-valued independently distributed ran∞ (i.i.d.) ∞ identical h i s i . f (s) = i=0 pi s i is the dom variables with generating function h(s) = i=0 probability generating function of the number of offspring per individual contributing to the next generation. ∞ i h i < ∞, p0 = Throughout this paper, we assume that h 0 < 1, 0 < a = i=0 ∞ 0, p j = 1 for any j ≥ 1, 1 < m = j=0 j p j < ∞. If h j = 0 for all j ≥ 0, i.e., there is no immigration, then {X n }∞ 0 is just the Galton– . Athreya [5] considered large deviation rates for Watson branching processes {Z n }∞ 0 and obtained the rates of convergence to 0 as n → ∞ of {Z n }∞ 0 Z n+1 − m > ε , P(|Wn − W | > ε), P Zn Z n+1 − m > ε|W ≥ α P Zn −n Z for ε > 0 and α > 0 under various moment conditions on { p j }∞ n 0 , where Wn = m and W is the limit of Wn in sense of almost surely. Bansaye and Berestycki [6] further considered large deviation for branching processes in a random environment. In the case that h
Data Loading...
