The fixed points of the multivariate smoothing transform

Abstract

A. Let be given a sequence (T 1 , T 2 , . . . ) of random d × d matrices with nonnegative entries and a random vector Q with nonnegative entries. Consider random vectors X with nonnegative entries, satisfyingwhere L = denotes equality of the corresponding laws, (X i ) i≥1 are i.i.d. copies of X and independent of (Q, T 1 , T 2 , . . . ).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. K: Smoothing transform, Markov random walks, general branching processes, multivariate stable laws, multitype branching random walk, Choquet-Deny lemma, weighted branching MSC: 60G44, 60E05, 60B20, 60B15be a sequence of random d × d-matrices and Q a random vector in R d . Assume that N := #{i : T i = 0} is finite a.s. We will presuppose throughout that the (T i ) i≥1 are ordered in such a way that T i = 0 if and only if i ≤ N . For any random variable X ∈ R d , let (X i ) i≥1 be a sequence of i.i.d. copies of X, and independent of (Q, (T i ) i≥1 ). Then the random variable N i=1 T i X i + Q is well defined, and one can ask the question, whether it holds thatwhere L = denotes equality in law. If X is a random variable such that (1.1) holds, then we call its law L (X) a solution of Eq. (1.1), and ask for a characterization of all solutions of Eq. (1.1). In other words, introducing the mapping S Q on the set P(R d ) of probability measures on R d , defined bySebastian Mentemeier. Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4,

Fixed points of the multivariate smoothing transform: the critical case

“…This shows in particular, that D(u 0 , t) is slowly varying as t → ∞. We then use the results of [27] to deduce that this already implies that D(u, t) is slowly varying for all u ∈ S ≥ . Lemma 4.1.…”

“…Moreover, if ψ(r) = E e −rX is the Laplace transform of a fixed point, then there is a positive function L, slowly varying at 0, and K > 0 such that Existence and uniqueness results in the multivariate setting d ≥ 2 for the non-critical case have been recently proved in [27]. The aim of this note is to provide the corresponding result for the multivariate critical case.…”

“…The aim of this note is to provide the corresponding result for the multivariate critical case. In order to so, we will first review necessary notation and definitions from [27], in particular introducing the multivariate analogue of the function m, as well as a result about the existence of fixed points in the critical case. Following the approach in [9,10,25] we will then prove that a multivariate regular variation property similar to (1.2) holds for fixed points (with an essentially unique, but yet undetermined slowly varying function L), which we use in order to prove the uniqueness of fixed points, up to scalars.…”

Generalized PageRank on directed configuration networks

“…The classification of their multiple solutions, calculation of their moments, and description of their asymptotic behavior, have been extensively studied in the literature, and are related to the broader study of weighted branching processes (WBPs) [60][61][62]. For the smoothing transform in particular, we mention the work in [1,2,4,5,24,32,39,[43][44][45]72] for the univariate case, and [8,23,41,51,52] for multivariate generalizations. The non-branching linear SFPE, i.e., N ≡ 1 in (1.4), is known as the random difference equation, and its analysis is even older, with some of the classical results being those found in [20,36,48].…”

Solutions to complex smoothing equations

“…have positive entries only and attention is restricted to solutions X on [0, ∞) d [64]. Yet, these results either cannot be applied to the complex case [64] or cover only a very special situation [14]. We will solve (1.1) in a multivariate setup that comfortably covers the complex case and thereby address open questions in papers by Barral [10,Remark 4] (posed for C = 0 and T 1 , T 2 , .…”