Bernoulli

  • Bernoulli
  • Volume 21, Number 1 (2015), 489-504.

Precise tail asymptotics of fixed points of the smoothing transform with general weights

D. Buraczewski, E. Damek, and J. Zienkiewicz

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Abstract

We consider solutions of the stochastic equation $R=_{d}\sum_{i=1}^{N}A_{i}R_{i}+B$, 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 $K$.

Article information

Source
Bernoulli Volume 21, Number 1 (2015), 489-504.

Dates
First available in Project Euclid: 17 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.bj/1426597079

Digital Object Identifier
doi:10.3150/13-BEJ576

Mathematical Reviews number (MathSciNet)
MR3322328

Zentralblatt MATH identifier
1321.60046

Keywords
large deviations linear stochastic equation regular variation smoothing transform

Citation

Buraczewski, D.; Damek, E.; Zienkiewicz, J. Precise tail asymptotics of fixed points of the smoothing transform with general weights. Bernoulli 21 (2015), no. 1, 489--504. doi:10.3150/13-BEJ576. https://projecteuclid.org/euclid.bj/1426597079


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