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How big is the minimum of a branching random walk?

Abstract : 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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https://hal.archives-ouvertes.fr/hal-00826652
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Submitted on : Sunday, July 2, 2017 - 5:14:58 PM
Last modification on : Wednesday, October 27, 2021 - 3:55:25 PM
Long-term archiving on: : Thursday, December 14, 2017 - 8:36:28 PM

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  • HAL Id : hal-00826652, version 5
  • ARXIV : 1305.6448

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Yueyun Hu. How big is the minimum of a branching random walk?. 2014. ⟨hal-00826652v5⟩

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