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Asymptotic behaviour of heavy-tailed branching processes in random environments. (English) Zbl 1466.60178
Summary: Consider a heavy-tailed branching process (denoted by $$Z_{n}$$) in random environments, under the condition which infers that $$\mathbb{E} \log m(\xi _{0})=\infty$$. We show that (1) there exists no proper $$c_{n}$$ such that $$\{Z_{n}/c_{n}\}$$ has a proper, non-degenerate limit; (2) normalized by a sequence of functions, a proper limit can be obtained, i.e., $$y_{n} (\bar{\xi} ,Z_{n}(\bar{\xi}))$$ converges almost surely to a random variable $$Y(\bar{\xi})$$, where $$Y\in (0,1)$$ $$\eta$$-a.s.; (3) finally, we give the necessary and sufficient conditions for the almost sure convergence of $$\{\frac{U(\bar {\xi},Z_{n}(\bar {\xi}))} {c_{n}(\bar{\xi})}\}$$, where $$U(\bar{\xi})$$ is a slowly varying function that may depend on $$\bar{\xi}$$.
##### MSC:
 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 60F10 Large deviations
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##### References:
 [1] Amini, O., Devroye, L., Griffiths, S. and Olver, N. (2013) On explosions in heavy-tailed branching random walks. Ann. Prob. 41(3B), 1864-1899. · Zbl 1304.60093 [2] Athreya, K.B. and Kailin, S. (1971) On branching processes with random environments I: Extinction probabilities. Ann. Math. Statist. 42, 1499-1520. · Zbl 0228.60032 [3] Athreya, K.B. and Kailin, S. (1971) On branching processes with random environments II: Limit theorems. Ann. Math. Statist. 42, 1843-1858. · Zbl 0228.60033 [4] Athreya, K.B. and Ney, P.E. (1972) Branching Processes.Springer, Berlin. · Zbl 0259.60002 [5] Cohn, H. (1977) Almost sure convergence of branching processes. Z. Wahrscheinlichkeitsth. 38, 73-81. · Zbl 0336.60074 [6] Darling, D.A. (1970) The Galton-Wstson process with infinite mean. J. Appl. Prob. 7, 455-456. · Zbl 0201.19302 [7] Grey, D.R. (1977) Almost sure convergence in Markov branching processes with infinite mean. J. Appl. Prob. 14, 702-716. · Zbl 0378.60064 [8] Heyde, C.C. (1970) Extension of a result of Seneta for the supercritical Galton-Watson process. Ann. Math. Statist. 41, 739-742. · Zbl 0195.19201 [9] Kesten, H. and Stigum, B.P. (1966) A limit theorem for multidimensional Galton-Watson processes. Ann. Math. Statist. 37, 1211-1223. · Zbl 0203.17401 [10] Schuh, H.-J. and Barbour, A.D. (1977) On the asymptotic behaviour of branching processes with infinite mean. Adv. Appl. Prob. 9, 681-723. · Zbl 0391.60083 [11] Seneta, E. (1969) Functional equations and the Galton-Watson process. Adv. Appl. Prob. 1, 1-42. · Zbl 0183.46105 [12] Seneta, E. (1973) The simple branching process with infinite mean I. J. Appl. Prob. 10, 206-212. · Zbl 0258.60058 [13] Tanny, D. (1977) Limit theorems for branching processes in a random environment. Ann. Prob. 5, 100-116. · Zbl 0368.60094 [14] Tanny, D. (1978) Normalizing constants for branching processes in random environments (B.P.R.E.). Stoch. Proc. and Their Appl. 6, 201-211. · Zbl 0371.60099 [15] Tanny, D. (1988) A necessary and sufficient condition for a branching process in a random environment to grow like the product of its means. Stoch. Proc. and Their Appl. 28, 123-139. · Zbl 0643.60064 [16] Vatutin, V. A. (1987) Sufficient conditions for the regularity of Bellman-Harris branching processes. Theory Probab. Appl. 31, 50-57. · Zbl 0658.60115
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