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July, 1992 Limit Theorems for the Frontier of a One-Dimensional Branching Diffusion
S. Lalley, T. Sellke
Ann. Probab. 20(3): 1310-1340 (July, 1992). DOI: 10.1214/aop/1176989693

Abstract

Let $R_t$ be the position of the rightmost particle at time $t$ in a time-homogeneous one-dimensional branching diffusion process. Let $\gamma(\alpha,t)$ be the $\alpha$th quantile of $R_t$ under $P^0$, where $P^x$ denotes the probability measure of the branching diffusion process starting with a single particle at position $x$. We show that $\gamma(\alpha,t)$ is a limiting quantile of $R_t$ under $P^x$ in the sense that $\lim_{t \rightarrow\infty}P^x\{R_t \leq \gamma(\alpha,t)\}$ exists for all $\alpha \in (0,1)$ and all $x \in \mathbb{R}$. If the underlying diffusion is recurrent, we show that, after an appropriate rescaling of space, the $P^x$ distribution of $R_t - t$ converges weakly to a nontrivial limiting distribution $w_x$.

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S. Lalley. T. Sellke. "Limit Theorems for the Frontier of a One-Dimensional Branching Diffusion." Ann. Probab. 20 (3) 1310 - 1340, July, 1992. https://doi.org/10.1214/aop/1176989693

Information

Published: July, 1992
First available in Project Euclid: 19 April 2007

zbMATH: 0759.60087
MathSciNet: MR1175264
Digital Object Identifier: 10.1214/aop/1176989693

Subjects:
Primary: 60J80
Secondary: 60F05 , 60G55

Keywords: branching diffusion process , extreme value distribution , Travelling wave

Rights: Copyright © 1992 Institute of Mathematical Statistics

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Vol.20 • No. 3 • July, 1992
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