Abstract
We consider a branching random walk in an independent and identically distributed random environment ξ = (ξn) indexed by the time. Let W be the limit of the martingale \(W_n=\int\;e^{-tx}Z_n(\text{d}x)/\mathbb{E}_\xi\int\;e^{-tx}Z_n(\text{d}x)\), with Zn denoting the counting measure of particles of generation n, and \(\mathbb{E}_\xi\) the conditional expectation given the environment ξ. We find necessary and sufficient conditions for the existence of quenched moments and weighted moments of W, when W is non-degenerate.
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Wang Y, Li Y, Liu Q, Liu Z. Quenched weighted moments of a supercritical branching process in a random environment. Published in Asian Journal of Mathematics, 2019
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The work has benefited from the support of the French government Investissements d’Avenir program ANR-11-LABX-0020-01. It has been partially supported by the National Natural Science Foundation of China (11571052, 11401590, 11731012 and 11671404), and by Hunan Natural Science Foundation (2017JJ2271).
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Wang, Y., Liu, Z., Liu, Q. et al. Asymptotic Properties of a Branching Random Walk with a Random Environment in Time. Acta Math Sci 39, 1345–1362 (2019). https://doi.org/10.1007/s10473-019-0513-y
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DOI: https://doi.org/10.1007/s10473-019-0513-y