Frontiers of Mathematics in China >
Fixed points of smoothing transformation in random environment
Received date: 09 Dec 2020
Accepted date: 06 Mar 2021
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At each time be a random sequence of non-negative numbers that are ultimately zero in a random environment. 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 ,where are random variables in random environment which satisfy that for any environment; under ; are independent of each other and , and have the same conditional distribution 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.
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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