Fixed points of smoothing transformation in random environment |
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Authors: | Xiaoyue ZHANG Wenming HONG |
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Affiliation: | 1. School of Statistics, Capital University of Economics and Business, Beijing 100070, China2. School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Beijing Normal University, Beijing 100875, China |
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Abstract: | 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. |
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Keywords: | Smoothing transformation functional equation branching random walk random environment martingales |
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