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Uniform convergence of martingales in the branching random walk. (English) Zbl 0748.60080

In a discrete-time supercritical branching random walk in \({\mathcal R}^ p\), let \(Z^{(n)}\) be the point process formed by the \(n\)th generation. Let \({\mathcal F}^{(n)}\) be the \(\sigma\)-field containing all information about the first \(n\) generations. Let \(\mu\) be the intensity measure of the \(Z^{(1)}\). Let \(m\) be the Laplace transform of \(\mu\). The author studies the convergence of the martingales \[ W^{(n)}(\lambda) = m(\lambda)^{-n}\int e^{-\lambda x}Z^{(n)}(dx). \] He gives first sufficient conditions for the almost sure convergence and \(L^{\alpha}\)- convergence, \(1<\alpha<2\), of the sequence \(W^{(n)}(\lambda)\). After this he can describe the set, where almost sure convergence and convergence in the mean is uniform with respect to \(\lambda\). Then he gives some large deviation results for \(Z^{(n)}\). At the end he gives continuous-time analogues for the above mentioned convergence and large deviations results.

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60G44 Martingales with continuous parameter
60F10 Large deviations
60G42 Martingales with discrete parameter
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