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The extremal process of branching Brownian motion. (English) Zbl 1286.60045
For a branching Brownian motion in which the offspring number has mean two and a finite variance, let \(n(t)\) be the number of particles which exist at time \(t>0\). Denote by \(X_k(t)\), \(1\leq k\leq n(t)\), the positions of these particles and set \(m(t):=\sqrt{2}t-{3\over 2\sqrt{2}}\log t\). It is known that \(\max_{1\leq k\leq n(t)}X_k(t)-m(t)\) converges in distribution as \(t\to\infty\). The authors prove that the point process \(\sum_{k\leq n(t)}\delta_{X_k(t)-m(t)}\) converges in distribution as \(t\to\infty\) to a Poisson cluster point process and provide a detailed description of the limit process. As the authors state in the abstract: “The proof combines three main ingredients. First, the results of Bramson on the convergence of solutions of the Kolmogorov-Petrovsky-Piscounov equation with general initial conditions to standing waves. Second, the integral representations of such waves as first obtained by Lalley and Sellke in the case of Heaviside initial conditions. Third, a proper identification of the tail of the extremal process with an auxiliary process (based on the work of B. Chauvin and A. Rouault [Math. Nachr. 149, 41–59 (1990; Zbl 0724.60091)]), which fully captures the large time asymptotics of the extremal process.”

MSC:
60G70 Extreme value theory; extremal stochastic processes
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
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