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

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 $K$.

Published at http://dx.doi.org/10.3150/13-BEJ576 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

References in corpus (2)

• Minima in branching random walks
• Heavy tailed solutions of multivariate smoothing transforms

Cited by in corpus (2)

• Heavy tailed solutions of multivariate smoothing transforms
• Large deviation estimates for exceedance times of perpetuity sequences and their dual processes