## 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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