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Galton-Watson branching processes in a random environment. I: Limit theorems. (English. Russian original) Zbl 1079.60080
Theory Probab. Appl. 48, No. 2, 314-336 (2003); translation from Teor. Veroyatn. Primen. 48, No. 2, 274-300 (2003).
Let $$Z_n= \{Z_n: n\geq 0\}$$ denote a Galton-Watson process in an i.i.d. random environment described by the probability generating functions $$f_0,f_1,\dots$$ . Set $$X_n:= \log f_{n-1}'(1)$$ and denote by $$S= \{S_n: n\geq 0\}$$ the random walk generated by the $$X_n$$. If $$Z$$ is “critical” in a sense, then $$\exp[-\min_{0\leq j\leq n}\,S_j]{\mathbf P}\{Z_n> 0\mid f_0,\dots, f_{n-1}\}$$ converges in law as $$n\uparrow\infty$$ to a positive finite random variable. Additionally, for a “typical” realization of the environment, $$\{Z_n\mid Z_n> 0\}$$ grows as $$\exp[S_n- \min_{0\leq j\leq n} S_j]$$ (except a constant factor).

##### MSC:
 60K37 Processes in random environments 60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
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