Regular variation of fixed points of the smoothing transform
Résumé
Let $(N,A_1,A_2,\ldots)$ be a sequence of random variables with $N\in \mathbb{N}\cup\{\infty\}$ and $A_i\in \mathbb{R}_+$. We are interested in asymptotic properties of non-negative solutions of the distributional equation
$Z=\sum_{i=1}^N A_i Z_i$, where $Z_i$ are nonnegative random variables independent of each other and independent of $(N,A_1,A_2,\ldots)$, each has the same distribution as $Z$ which is unknown. For a solution $Z$ with finite mean, we show that under a natural moment condition, the regular variation of $P(Z>x)$ $ (x\rightarrow \infty)$ is equivalent to that of $P(Y_1>x)$, where $Y_1=\sum_{i=1}^N A_i$. The results generalize the corresponding theorems of Bingham and Doney (1974, 1975) on the Galton-Watson process and the Crump-Mode-Jirina process, and improve those of Iksanov and Polotskiy (2006) on the branching random walk.
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