We consider a branching random walk on $${\mathbb {R}}$$ R with a stationary and ergodic environment $$\xi =(\xi _n)$$ ξ = ( ξ n ) indexed by time $$n\in {\mathbb {N}}$$ n ∈ N . Let $$Z_n$$ Z n be the counting measure of particles of generation n and $$\tilde{Z}_n(t)=\int \mathrm{e}^{tx}Z_n(\mathrm{d}x)$$ Z ~ n ( t ) = ∫ e t x Z n ( d x ) be its Laplace transform. We show the $$L^p$$ L p convergence rate and the uniform convergence of the martingale $$\tilde{Z}_n(t)/{\mathbb {E}}[\tilde{Z}_n(t)|\xi ]$$ Z ~ n ( t ) / E [ Z ~ n ( t ) | ξ ] , and establish a moderate deviation principle for the measures $$Z_n$$ Z n .
Journal of Theoretical Probability – Springer Journals
Published: Jan 23, 2016
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