RESEARCH ARTICLE

Fixed points of smoothing transformation in random environment

  • Xiaoyue ZHANG 1 ,
  • Wenming HONG , 2
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  • 1. School of Statistics, Capital University of Economics and Business, Beijing 100070, China
  • 2. School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Beijing Normal University, Beijing 100875, China

Received date: 09 Dec 2020

Accepted date: 06 Mar 2021

Copyright

2021 Higher Education Press

Abstract

At each time nN,letY¯(n)(ξ)=(y1(n)(ξ),y2(n)(ξ),) be a random sequence of non-negative numbers that are ultimately zero in a random environmentξ=ξnnN. The existence and uniqueness of the nonnegative fixed points of the associated smoothing transformation in random environment are considered. These fixed points are solutions to the distributional equation for a.e.ξ,Z(ξ)=di+yi(0)(ξ)Zi(1)(ξ),where Zi(1):i+ are random variables in random environment which satisfy that for any environmentξ; under Pξ; Zi(1):i+are independent of each other and Y(0)(ξ), and have the same conditional distribution Pξ(Zi(1)(ξ))=PTξ(Z(Tξ)) 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.

Cite this article

Xiaoyue ZHANG , Wenming HONG . Fixed points of smoothing transformation in random environment[J]. Frontiers of Mathematics in China, 2021 , 16(4) : 1191 -1210 . DOI: 10.1007/s11464-021-0934-5

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