Central Limit Theorems for Super Ornstein-Uhlenbeck Processes

作者：Ren Yan Xia; Song Renming; Zhang Rui^*

来源：Acta Applicandae Mathematicae, 2014, 130(1): 9-49.

DOI：10.1007/s10440-013-9837-0

• Central limit theorem
• Backbone decomposition
• Superprocess
• Super Ornstein-Uhlenbeck process
• Branching process
• Branching Ornstein-Uhlenbeck process
• Ornstein-Uhlenbeck process

Suppose that X={X-t : t >= 0} is a supercritical super Ornstein-Uhlenbeck process, that is, a superprocess with an Ornstein-Uhlenbeck process on R-d corresponding to L = 1/2 sigma(2)Delta - bx . del as its underlying spatial motion and with branching mechanism psi(lambda)=-alpha lambda+beta lambda(2)+integral((0,+infinity))(e(-lambda x) -1+lambda x)n(dx), where alpha=-psi'(0+)> 0, beta >= 0, and n is a measure on (0, infinity) such that integral((0,+infinity)) x(2)n(dx)<+infinity. Let P-mu be the law of X with initial measure mu. Then the process W-t =e(-alpha t) parallel to X-t parallel to is a positive P-mu-martingale. Therefore there is W-infinity such that W-t -> W infinity, P-mu-a.s. as t -> infinity. In this paper we establish some spatial central limit theorems for X.
Let P denote the function class
P := {f is an element of C(R-d) : there exists k is an element of N such that vertical bar f(x)vertical bar/parallel to x parallel to(k) -> 0 as parallel to x parallel to -> infinity}.
For each f is an element of P we define an integer gamma(f) in term of the spectral decomposition of f. In the small branching rate case alpha < 2 gamma(f)b, we prove that there is constant sigma(2)(f) is an element of (0, infinity) such that, conditioned on no-extinction,
(e(-alpha t) parallel to X-t parallel to, < f, X-t >/root parallel to X parallel to) ->(d) (W*, G(1)(f)), t -> infinity,
where W* has the same distribution as W-infinity conditioned on no-extinction and G(1)(f) similar to N(0, sigma(2)(f)). Moreover, W* and G(1)(f) are independent. In the critical rate case alpha = 2 gamma(f)b, we prove that there is constant rho(2)(f) is an element of (0, infinity) such that, conditioned on no-extinction,
(e(-alpha t) parallel to X-t parallel to, < f, X-t >/t(1/2) root parallel to X-t parallel to) ->(d) (W*, G(2)(f)), t -> infinity,
where W* has the same distribution as W-infinity conditioned on no-extinction and G(2)(f) similar to N(0, rho(2)(f)). Moreover W* and G(2)(f) are independent. We also establish two central limit theorems in the large branching rate case alpha > 2 gamma(f)b.
Our central limit theorems in the small and critical branching rate cases sharpen the corresponding results in the recent preprint of Milos in that our limit normal random variables are non-degenerate. Our central limit theorems in the large branching rate case have no counterparts in the recent preprint of Milos. The main ideas for proving the central limit theorems are inspired by the arguments in K. Athreya's 3 papers on central limit theorems for continuous time multi-type branching processes published in the late 1960's and early 1970's.

• 出版日期2014-4
• 单位北京大学