Precise Tail Index of Fixed Points of the Two-Sided Smoothing Transform

Abstract

We consider real-valued random variables R satisfying the distributional equation

$$\displaystyle{ R\stackrel{d}{=}\sum _{k=1}^{N}T_{ k}R_{k} + Q, }$$

where $R_{1},R_{2},\ldots$ are iid copies of R and independent of $\mathbf{T} = (Q,(T_{k})_{k\geq 1})$. N is the number of nonzero weights T _k and assumed to be a.s. finite. Its properties are governed by the function

$$\displaystyle{m(s):= \mathbb{E}\sum _{k=1}^{N}{\left \vert T_{ k}\right \vert }^{s}.}$$

There are at most two values α < β such that $m(\alpha ) = m(\beta ) = 1$. We consider solutions R with finite moment of order s > α. We review results about existence and uniqueness. Assuming the existence of β and an additional mild moment condition on the T _k , our main result asserts that

$$\displaystyle{\lim _{t\rightarrow \infty }{t}^{\beta }\mathbb{P}(\left \vert R\right \vert > t)\ =\ K\ >\ 0.}$$

the main contribution being that K is indeed positive and therefore β the precise tail index of | R | , for the convergence was recently shown by Jelenkovic and Olvera-Cravioto [10].