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

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).