The fixed points of the multivariate smoothing transform

Abstract

Let (T1,T2,…) be a sequence of random d×d matrices with nonnegative entries, and let Q be a random vector with nonnegative entries. Consider random vectors X with nonnegative entries, satisfying X=L∑i≥1TiXi+Q,where =L denotes equality of the corresponding laws, (Xi)i≥1 are i.i.d. copies of X and independent of (Q,T1,T2,…). For d=1, this equation, known as fixed point equation of the smoothing transform, has been intensively studied. Under assumptions similar to the one-dimensional case, we obtain a complete characterization of all solutions X to (*) in the non-critical case, and existence results in the critical case.