The fixed points of the multivariate smoothing transform

2013/09/30 by Sebastian Mentemeier · 14 citations
Mathematics · #Fixed point #Holomorphic and Operator Theory #Mathematical finance #Multivariate random variable #Multivariate statistics #Point processes and geometric inequalities #Random Matrices and Applications #Random variable #Sequence (biology) #Smoothing #math.PR #msc:60B15 #msc:60B20 #msc:60E05 #msc:60G44

paper · pdf · doi:10.1007/s00440-015-0615-y

published in Probability Theory and Related Fields 164(1-2), 401-458 (Springer Science+Business Media) · 48 pages

arxiv created 2014/11/24 · arxiv updated 2015/01/09 · openalex publication_date 2015/01/20 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

Let N,d > 1 be fixed integers, let (T₁, ..., T_N) be random d-by-d matrices with nonnegative entries and Q a random d-vector with nonnegative entries. This induces a mapping (the multivariate smoothing transform) on probability laws on the nonnegative cone by S η:= Law of (T₁ X₁ + ... + T_N X_N + Q), where the X_i are iid with law η and independent of (T₁, ..., T_N, Q). Under conditions similar to those for the well-studied case d=1, a complete characterization of all fixed points of S is obtained.

Citations

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