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

Abstract

AbstractWe 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].KeywordsJelenkovicFinite MomentsMain Result AssertsReal-valued Random VariablesNonzero WeightsThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.