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Asymptotic behaviour of critical controlled branching processes with random control functions. (English) Zbl 1079.60073
Let $$Z_{n+1} = \sum^{\varphi_n (Z_n)} X_{n, j}$$ be a controlled branching process with i.i.d. random control functions $$\varphi_n$$. In the critical case, $$E(Z_{n+1}/Z_n\mid Z_n = k) \rightarrow_k 1$$, the authors establish results on extinction, nonextinction and of the limiting distribution under suitable normalization. The main tool are methods for growth processes $$Z_{n+1} = Z_n + g (Z_n) \xi_n$$ as in [G. Keller, G. Kersting and the reviewer, Ann. Prob. 15, 305–343 (1987; Zbl 0616.60079)].
Reviewer: Uwe Rösler (Kiel)

##### MSC:
 60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
##### Keywords:
extinction; weak limit; growth process
Full Text:
##### References:
 [1] Bruss, F. T. (1980). A counterpart of the Borel–Cantelli lemma. J. Appl. Prob. 17 , 1094–1101. JSTOR: · Zbl 0443.60002 [2] Chow, Y. S. and Teicher, H. (1997). Probability Theory. Independence, Interchangeability, Martingales, 3rd edn. Springer, New York. · Zbl 0891.60002 [3] Dion, J.-P. and Essebbar, B. (1995). On the statistics of controlled branching processes. In Branching Processes (Lecture Notes Statist. 99 ), ed. C. C. Heyde, Springer, New York, pp. 14–21. · Zbl 0821.62046 [4] González, M., Molina, M. and del Puerto, I. (2002). On the controlled Galton–Watson process with random control function. J. Appl. Prob. 39 , 804–815. · Zbl 1032.60077 [5] González, M., Molina, M. and del Puerto, I. (2003). On the geometric growth in controlled branching processes with random control function. J. Appl. Prob. 40 , 995–1006. · Zbl 1054.60087 [6] González, M., Molina, M. and del Puerto, I. (2004). Limiting distribution for subcritical controlled branching processes with random control function. Statist. Prob. Lett. 67 , 277–284. · Zbl 1063.60121 [7] González, M., Molina, M. and del Puerto, I. (2005). On $$L^2$$-convergence of controlled branching processes with random control function. Bernoulli 11 , 37–46. · Zbl 1062.60088 [8] Höpfner, R. (1985). On some classes of population-size-dependent Galton–Watson processes. J. Appl. Prob. 22 , 25–36. JSTOR: · Zbl 0573.60079 [9] Höpfner, R. (1986). Some results on population-size-dependent Galton–Watson processes. J. Appl. Prob. 23 , 297–306. JSTOR: · Zbl 0598.60089 [10] Jagers, P. (1975). Branching Processes with Biological Applications . John Wiley, London. · Zbl 0356.60039 [11] Keller, G., Kersting, G. and Rösler, U. (1987). On the asymptotic behaviour of discrete time stochastic growth processes. Ann. Prob. 15 , 305–343. · Zbl 0616.60079 [12] Kersting, G. (1986). On recurrence and transience of growth models. J. Appl. Prob. 23 , 614–625. JSTOR: · Zbl 0611.60084 [13] Kersting, G. (1992). Asymptotic $$\Gamma$$-distribution for stochastic difference equations. Stoch. Process. Appl. 40 , 15–28. · Zbl 0747.60024 [14] Klebaner, F. (1989). Stochastic difference equations and generalized gamma distributions. Ann. Prob. 17 , 178–188. · Zbl 0674.60077 [15] Nakagawa, T. (1994). The $$L^\alpha$$ ($$1<\alpha\leq2$$) convergence of a controlled branching process in a random environment. Bull. Gen. Ed. Dokkyo Univ. School Medicine 17 , 17–24. [16] Yanev, G. P. and Yanev, N. M. (1995). Critical branching process with random migration. In Branching Processes (Lecture Notes Statist. 99 ), ed. C. C. Heyde, Springer, New York, pp. 36–46. · Zbl 0833.60086 [17] Yanev, G. P. and Yanev, N. M. (2004). A critical branching process with stationary-limiting distribution. Stoch. Anal. Appl. 22 , 721–738. · Zbl 1085.60062 [18] Yanev, G. P., Mitov, K. V. and Yanev, N. M. (2003). Critical branching regenerative process with random migration. J. Appl. Statist. Sci. 12 , 41–54. · Zbl 1052.60071 [19] Yanev, N. M. (1976). Conditions for degeneracy of $$\phi$$-branching processes with random $$\phi$$. Theory Prob. Appl. 20 , 421–428. · Zbl 0363.60072
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