How big is the minimum of a branching random walk?
Résumé
Let $\M_n$ be the minimal position at generation $n$, of a real-valued branching random walk in the boundary case. As $n \to \infty$, $\M_n- {3 \over 2} \log n$ is tight (see \cite{1, 9, 2}). We establish here a law of iterated logarithm for the upper limits of $\M_n$: upon the system's non-extinction, $ \limsup_{n\to \infty} {1\over \log \log \log n} ( \M_n - {3\over2} \log n) = 1$ almost surely. We also study the problem of moderate deviations of $\M_n$: $\p(\M_n- {3 \over 2} \log n > \lambda)$ for $\lambda\to \infty$ and $\lambda=o(\log n)$. This problem is closely related to the small deviations of a class of Mandelbrot's cascades.
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