Precise tail index of fixed points of the two-sided smoothing transform

TL;DR

This paper investigates the tail behavior of solutions to a stochastic fixed point equation, establishing the exact tail index and showing the positivity of the tail constant under mild conditions.

Contribution

It proves that the tail index of the solution's distribution is precisely the value where the moment function equals one, confirming the tail constant is positive.

Findings

The tail probability decays as t^(-β) with a positive constant K.

The tail index β is exactly the solution to m(β)=1.

The results extend previous work by confirming the positivity of the tail constant.

Abstract

We consider real-valued random variables R satisfying the distributional equation R \eqdist \sum_{k=1}^{N}T_k R_k + Q, where R_1,R_2,... are iid copies of R and independent of T=(Q, (T_k)_{k \ge 1}). N is the number of nonzero weights T_k and assumed to be a.s. finite. Its properties are governed by the function m(s) := \E \sum_{k=1}^N |T_k|^s . There are at most two values \alpha < \beta such that m(\alpha)=m(\beta)=1. We consider solutions R with finite moment of order s > \alpha. We review results about existence and uniqueness. Assuming the existence of \beta and an additional mild moment condition on the T_{k}, our main result asserts that \lim_{t \to \infty} t^\beta P(|R| > t) = K > 0, the main contribution being that K is indeed positive and therefore \beta the precise tail index of |R|, for the convergence was recently shown by Jelenkovic and Olvera-Cravioto (arXiv:1012.2165).