Fixed points of smoothing transformation in random environment

摘要

At each time n is an element of N, let (Y) over bar (n)(xi)=(y(1)((n))(xi),y(2)((n))(xi)...), be a random sequence of non-negative numbers that are ultimately zero in a random environment xi = (xi(n))(n is an element of N). The existence and uniqueness of the non-negative fixed points of the associated smoothing transformation in random environment are considered. These fixed points are solutions to the distributional equation for a.e. xi,Z(xi)=d Sigma(i is an element of N)+y(i)((0))(xi)Z(i)(1)(xi), where {Zi(1):i is an element of N+}documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} are random variables in random environment which satisfy that for any environment xi, under P-xi, {Zi(1)(xi):i is an element of N+} are independent of each other and (Y) over bar ((0))(xi), and have the same conditional distribution P xi(Zi(1)(xi)is an element of.)=PT xi(Z(T xi)is an element of where T is the shift operator. This extends the classical results of J. D. Biggins [J. Appl. Probab., 1977, 14: 25-37] to the random environment case. As an application, the martingale convergence of the branching random walk in random environment is given as well.