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Precise tail asymptotics of fixed points of the smoothing transform with general weights (1504.03144v1)
Published 13 Apr 2015 in math.ST and stat.TH
Abstract: We consider solutions of the stochastic equation $R=d\sum{i=1}NA_iR_i+B$, where $N>1$ is a fixed constant, $A_i$ are independent, identically distributed random variables and $R_i$ are independent copies of $R$, which are independent both from $A_i$'s and $B$. The hypotheses ensuring existence of solutions are well known. Moreover under a number of assumptions the main being $\mathbb{E}|A_1|{\alpha}=1/N$ and $\mathbb{E}|A_1|{\alpha}\log|A_1|>0$, the limit $\lim_{t\to\infty}t{\alpha}\mathbb{P}[|R|>t]=K$ exists. In the present paper, we prove positivity of $K$.