Acta Applicandae Mathematicae

, Volume 130, Issue 1, pp 9–49

# Central Limit Theorems for Super Ornstein-Uhlenbeck Processes

Article

## Abstract

Suppose that X={Xt:t≥0} is a supercritical super Ornstein-Uhlenbeck process, that is, a superprocess with an Ornstein-Uhlenbeck process on $$\mathbb{R}^{d}$$ corresponding to $$L=\frac{1}{2}\sigma^{2}\Delta-b x\cdot\nabla$$ as its underlying spatial motion and with branching mechanism ψ(λ)=−αλ+βλ2+∫(0,+∞)(eλx−1+λx)n(dx), where α=−ψ′(0+)>0, β≥0, and n is a measure on (0,∞) such that ∫(0,+∞)x2n(dx)<+∞. Let $$\mathbb{P} _{\mu}$$ be the law of X with initial measure μ. Then the process Wt=eαtXt∥ is a positive $$\mathbb{P} _{\mu}$$-martingale. Therefore there is W such that WtW, $$\mathbb{P} _{\mu}$$-a.s. as t→∞. In this paper we establish some spatial central limit theorems for X.

Let $$\mathcal{P}$$ denote the function class
$$\mathcal{P}:=\bigl\{f\in C\bigl(\mathbb{R}^d\bigr): \mbox{there exists } k\in\mathbb{N} \mbox{ such that }|f(x)|/\|x\|^k\to 0 \mbox{ as }\|x\|\to\infty \bigr\}.$$
For each $$f\in\mathcal{P}$$ we define an integer γ(f) in term of the spectral decomposition of f. In the small branching rate case α<2γ(f)b, we prove that there is constant $$\sigma_{f}^{2}\in (0,\infty)$$ such that, conditioned on no-extinction,
\begin{aligned} \biggl(e^{-\alpha t}\|X_t\|, ~\frac{\langle f , X_t\rangle}{\sqrt{\|X_t\|}} \biggr) \stackrel{d}{\rightarrow}\bigl(W^*,~G_1(f)\bigr), \quad t\to\infty, \end{aligned}
where W has the same distribution as W conditioned on no-extinction and $$G_{1}(f)\sim \mathcal{N}(0,\sigma_{f}^{2})$$. Moreover, W and G1(f) are independent. In the critical rate case α=2γ(f)b, we prove that there is constant $$\rho_{f}^{2}\in (0,\infty)$$ such that, conditioned on no-extinction,
\begin{aligned} \biggl(e^{-\alpha t}\|X_t\|, ~\frac{\langle f , X_t\rangle}{t^{1/2}\sqrt{\|X_t\|}} \biggr) \stackrel{d}{\rightarrow}\bigl(W^*,~G_2(f)\bigr), \quad t\to\infty, \end{aligned}
where W has the same distribution as W conditioned on no-extinction and $$G_{2}(f)\sim \mathcal{N}(0, \rho_{f}^{2})$$. Moreover W and G2(f) are independent. We also establish two central limit theorems in the large branching rate case α>2γ(f)b.

Our central limit theorems in the small and critical branching rate cases sharpen the corresponding results in the recent preprint of Miłoś 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 Miłoś. 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.

### Keywords

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

### Mathematics Subject Classification (2000)

60J80 60G57 60J45

### References

1. 1.
Adamczak, R., Miłoś, P.: CLT for Ornstein-Uhlenbeck branching particle system. Preprint (2011). arXiv:1111.4559
2. 2.
Asmussen, S., Hering, H.: Strong limit theorems for general supercritical branching processes with applications to branching diffusions. Z. Wahrs. Verw. Gebiete 36(3), 195–212 (1976)
3. 3.
Asmussen, S., Hering, H.: Strong limit theorems for supercritical immigration branching processes. Math. Scand. 39(2), 327–342 (1977)
4. 4.
Asmussen, S., Hering, H.: Branching Processes. Birkhäuser, Boston (1983)
5. 5.
Asmussen, S., Keiding, N.: Martingale central limit theorems and asymptotic estimation theory for multitype branching processes. Adv. Appl. Probab. 10, 109–129 (1978)
6. 6.
Athreya, K.B.: Limit theorems for multitype continuous time Markov branching processes I: The case of an eigenvector linear functional. Z. Wahrs. Verw. Gebiete 12, 320–332 (1969)
7. 7.
Athreya, K.B.: Limit theorems for multitype continuous time Markov branching processes II: The case of an arbitrary linear functional. Z. Wahrs. Verw. Gebiete 13, 204–214 (1969)
8. 8.
Athreya, K.B.: Some refinements in the theory of supercritical multitype Markov branching processes. Z. Wahrs. Verw. Gebiete 20, 47–57 (1971)
9. 9.
Berestycki, J., Kyprianou, A.E., Murillo-Salas, A.: The prolific backbone for supercritical superprocesses. Stoch. Proc. Appl. 121, 1315–1331 (2011)
10. 10.
Conner, H.E.: Asymptotic behavior of averaging processes for a branching process of restricted Brownian particles. J. Math. Anal. Appl. 20, 464–479 (1967)
11. 11.
Davis, A.W.: Branching-diffusion processes with no absorbing boundaries, I. J. Math. Anal. Appl. 18, 276–296 (1967)
12. 12.
Dudley, R.M.: Real Analysis and Probability. Cambridge University Press, Cambridge (2002)
13. 13.
Dynkin, E.B.: Superprocesses and partial differential equations. Ann. Probab. 21, 1185–1262 (1993)
14. 14.
Dynkin, E.B., Kuznetsov, S.E.: $$\mathbb{N}$$-Measure for branching exit Markov system and their applications to differential equations. Probab. Theory Rel. Fields 130, 135–150 (2004)
15. 15.
El Karoui, N., Roelly, S.: Propriétés de martingales, explosion et représentation de Lévy-Khintchine d’une classe de processus de branchment à valeurs mesures. Stoch. Proc. Appl. 38, 239–266 (1991)
16. 16.
Engländer, J.: Law of large numbers for superdiffusions: the non-ergodic case. Ann. Inst. H. Poincaré Probab. Statist. 45, 1–6 (2009)
17. 17.
Engländer, J., Harris, S.C., Kyprianou, A.E.: Strong law of large numbers for branching diffusions. Ann. Inst. H. Poincaré Probab. Statist. 46, 279–298 (2010)
18. 18.
Engländer, J., Winter, A.: Law of large numbers for a class of superdiffusions. Ann. Inst. H. Poincaré Probab. Statist. 42, 171–185 (2006)
19. 19.
Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vol. II. McGraw-Hill, New York (1953)
20. 20.
Hardy, R., Harris, S.C.: A spine aproach to branching diffusions with applications to L p-convergence of martingales. In: Séminaire de Probabilités XLII. Lecture Notes in Math., vol. 1979, pp. 281–330 (2009)
21. 21.
Kesten, H., Stigum, B.P.: A limit theorem for multidimensional Galton-Watson processes. Ann. Math. Statist. 37, 1211–1223 (1966)
22. 22.
Kesten, H., Stigum, B.P.: Additional limit theorems for indecomposable multidimensional Galton-Watson processes. Ann. Math. Statist. 37, 1463–1481 (1966)
23. 23.
Kyprianou, A.E.: Introductory Lectures on Fluctuations of Levy Processes with Applications. Springer, Berlin (2006)
24. 24.
Li, Z.: Skew convolution semigroups and related immigration processes. Theory Probab. Appl. 46, 274–296 (2003)
25. 25.
Li, Z.: Measure-Valued Branching Markov Processes. Springer, Heidelberg (2011)
26. 26.
Liu, R., Ren, Y.-X., Song, R.: LlogL criterion for a class of superdiffusions. J. Appl. Probab. 46, 479–496 (2009)
27. 27.
Liu, R., Ren, Y.-X., Song, R.: Strong law of large numbers for a class of superdiffusions. Acta Appl. Math. 123, 73–97 (2013)
28. 28.
Metafune, G., Pallara, D.: Spectrum of Ornstein-Uhlenbeck operators in $$\mathcal{L}^{p}$$ spaces with respect to invariant measures. J. Funct. Anal. 196, 40–60 (2002)
29. 29.
Miłoś, P.: Spatial CLT for the supercritical Ornstein-Uhlenbeck superprocess. Preprint (2012). arXiv:1203.6661
30. 30.
Ren, Y.-X., Song, R., Zhang, R.: Central limit theorems for supercritical branching Markov processes. Preprint (2013). arXiv:1305.0610
31. 31.
Watanabe, S.: Limit theorem for a class of branching processes. In: 1967 Markov Processes and Potential Theory, Madison, Wis., 1967. Proc. Sympos. Math. Res. Center, pp. 205–232. Wiley, New York (1967) Google Scholar