Limit theorems for the minimal position in a branching random walk with independent logconcave displacements

By Markus Bachmann. Advances in Applied Probability, 2000

62 citations recorded by the source corpus.

Subfields: Artificial Intelligence, Mathematical Physics, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/d5ed03334e8edc15c18225ec74c9b9ef380b22cb

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Citations in the corpus

Maximum of the Characteristic Polynomial of Random Unitary Matrices (2015, unknown): “This started with the seminal work of Bramson [15] for branching Brownian motion and was later extended to general branching random walks [2,3,7,17,18].”
Extrema of log-correlated random variables: Principles and Examples (2016, background): “For branching random walks, it was proved in great generality by Aı̈dekon [3], see also [1,14,24]; for the two-dimensional Gaussian free field, by Bramson et al.”
Branching random walks (2015, background): “12 without getting the optimal value 3/2, and tightness in Bramson and Zeitouni [71]; also, convergence in a special case (including the example of Gaussian displacements), centered at the mean, is proved by Bachmann [33].”
A survey of max-type recursive distributional equations (2004, result): “We quote a recent result from [14], which provides an extensive bibliography of earlier work.”
Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees (2007, background): “See, for example, Bachmann [4] and Bramson and Zeitouni [14], as well as Section 5 of the survey paper by Aldous and Bandyopadhyay [2].”
Travelling wave solutions to the K-P-P equation: alternatives to Simon Harris' probabilistic analysis (2004, background): “Indeed embedded in the results of [5] is the existence of a Seneta–Heyde sequence for an analogue of the martingale W under anx logx moment condition and within the context of a spec class of branching random walk.”
Convergence in law of the minimum of a branching random walk (2013, background): “In the particular case where the step distribution is log-concave, the convergence in law of Mn around its median was proved earlier by Bachmann [4].”
Minima in branching random walks (2007, background): “Bachmann [3, 4] studied the tightness of the family {Mn − mn :n ∈ N}, where mn is the median of Mn, showing that the aforementioned family is tight when X has logarithmically concave distribution function.”
Convergence in law of the minimum of a branching random walk (2011, background): “In the particular case where the step distribution is log-concave, the convergence in law of Mn around its median was proved earlier by Bachmann [4].”
Convergence of the centered maximum of log-correlated Gaussian fields (2015, unknown): “Convergence for the centered maximum of BRW with Gaussian increments was obtained in [2], while the analogous result for general BRWs under mild assumptions was only obtained quite recently in the important work [1], using the notion of derivative martingale to describe the limit; see also [8].”