A random-environment version of the smoothing transform has unique non-negative finite-mean fixed points exactly under Biggins-type moment and drift conditions, and this yields the martingale convergence theorem for branching random walks in random environments.
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Fixed points with finite mean of the smoothing transform in random environments
A random-environment version of the smoothing transform has unique non-negative finite-mean fixed points exactly under Biggins-type moment and drift conditions, and this yields the martingale convergence theorem for branching random walks in random environments.