Asymptotic Properties of a Branching Random Walk with a Random Environment in Time
Résumé
We consider a branching random walk in an independent and identically distributed random environment $\xi =(\xi_n)$ indexed by the time. Let $W$ be the limit of the martingale $W_n = \int e^{-tx}Z_{n}(dx)/ \mathbb{E}_{\xi}\int e^{-tx}Z_{n}(dx)$, with $ Z_n $ denoting the counting measure of particles of generation $n$, and $\mathbb{E}_{\xi}$ the conditional expectation given the environment $\xi$. 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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