Fixed points of the smoothing transform; the boundary case
نویسنده
چکیده
Let A = (A1, A2, A3, . . .) be a random sequence of non-negative numbers that are ultimately zero with E[ ∑ Ai] = 1 and E [ ∑ Ai log Ai] ≤ 0. The uniqueness of the non-negative fixed points of the associated smoothing transform is considered. These fixed points are solutions to the functional equation Φ(ψ) = E [ ∏ i Φ(ψAi)] , where Φ is the Laplace transform of a non-negative random variable. The study complements, and extends, existing results on the case when E [ ∑ Ai log Ai] < 0. New results on the asymptotic behaviour of the solutions near zero in the boundary case, where E [ ∑ Ai log Ai] = 0, are obtained.
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