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On tails of fixed points of the smoothing transform in the boundary case. (English) Zbl 1183.60031
Let $$\{A_i\}$$ be a sequence of positive random numbers such that only the first $$N$$ of them are strictly positive, $$N$$ being an a.s. finite random integer. Consider the nonnegative solutions of the distribution equation $$Z=_d\sum^N_{i=1} A_i Z_i$$, where $$Z,Z_1,Z_2,\dots$$ are i.i.d. and independent of $$N,A_1,A_2,\dots$$ . If $${\mathbf E} \sum^N_{i= 1} A_i= 1$$ and $${\mathbf E}\sum^N_{i=1} A_i\log A_i= 0$$, it is known that solutions exist and that they all have infinite mean. Assuming in addition that $${\mathbf E}\sum^N_{i=1} A^{1-\delta}_i< \infty$$ for some $$\delta> 0$$, $${\mathbf E}(\sum^N_{i=1} JA_i)^{1+ \varepsilon}< \infty$$ for some $$\varepsilon> 0$$, and $$1<{\mathbf E} N\leq\infty$$, the author proves the following:
Let $$Z$$ be a nonnegative, non-zero solution. If the $$A_i$$ are aperiodic (i.e., there is no positive number $$h$$ such that $$\log A_i$$ is a.s. an integer multiple of $$h$$, for $$1\leq i\leq N$$), then $$\lim_{x\to\infty} x{\mathbf P}(Z> x)= C_0$$ for some finite strictly positive number $$C_0$$. If the $$A_i$$ are periodic, then there exist two positive constants $$C_1$$ and $$C_2$$ such that $$C_1=\liminf_{x\to\infty} x{\mathbf P}(Z> x)\leq \limsup_{x\to\infty} x{\mathbf P}(Z> x)= C_2$$. The problem is reduced to the study of the behaviour at infinity of an invariant measure of a random difference equation, see Y. Guivarc’h [Ann. Inst. Henri Poincaré, Probab. Stat. 26, No. 2, 261–285 (1990; Zbl 0703.60012)] and Q. Liu [Stochastic Processes Appl. 86, No. 2, 263–286 (2000; Zbl 1028.60087)].

##### MSC:
 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 60G42 Martingales with discrete parameter
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