Fixed points of multivariate smoothing transforms with scalar weights

Abstract

Given a sequence $(C_1,\dots ,C_d,T_1,T_2,\dots )$ of real-valued random variables with $N:=\#\{j\ge 1:T_j\ne 0\}<\infty$ almost surely, there is an associated smoothing transformation which maps a distribution $P$ on ${\mathbb{R}}^d$ to the distribution of ${\sum }_{j\ge 1}{T}_j{\mathbf{X}}^{(j)}+\mathbf{C}$ where $\mathbf{C}=(C_1,\dots ,C_d)$ and $({\mathbf{X}}^{(j)}{)}_{j\ge 1}$ is a sequence of independent random vectors with distribution $P$ independent of $(C_1,\dots ,C_d,T_1,T_2,\dots )$. We are interested in the fixed points of this mapping. By improving on the techniques developed in [G. Alsmeyer, J.D. Biggins, and M. Meiners. The functional equation of the smoothing transform {\em Ann. Probab.}, 40(5):2069--2105, 2012] and [G. Alsmeyer and M. Meiners. Fixed points of the smoothing transform: two-sided solutions. {\em Probab. Theory Related Fields}, 155(1-2):165--199, 2013], we determine the set of all fixed points under weak assumptions on $(C_1,\dots ,C_d,T_1,T_2,\dots )$. In contrast to earlier studies, this includes the most intricate case when the $T_j$ take both positive and negative values with positive probability. In this case, in some situations, the set of fixed points is a subset of the corresponding set when the $T_j$ are replaced by their absolute values, while in other situations, additional solutions arise.

43 pages

References in corpus (3)

• Heavy tailed solutions of multivariate smoothing transforms
• Rate of convergence in the law of large numbers for supercritical general multi-type branching processes
• Regular variation in the branching random walk

Cited by in corpus (6)

• Rate of convergence in the law of large numbers for supercritical general multi-type branching processes
• Continuity estimates for the complex cascade model on the phase boundary
• Coupling on weighted branching trees
• Convergence of complex martingales in the branching random walk: the boundary
• Solutions to complex smoothing equations
• Precise Tail Asymptotics for Attracting Fixed Points of Multivariate Smoothing Transformations