Electronic Journal of Probability

The Exact Asymptotic of the Time to Collision

Zbigniew Puchala and Tomasz Rolski

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In this note we consider the time of the collision $\tau$ for $n$ independent copies of Markov processes $X^1_t,. . .,X^n_t$, each starting from $x_i$,where $x_1 <. . .< x_n$. We show that for the continuous time random walk $P_{x}(\tau > t) = t^{-n(n-1)/4}(Ch(x)+o(1)),$ where $C$ is known and $h(x)$ is the Vandermonde determinant. From the proof one can see that the result also holds for $X_t$ being the Brownian motion or the Poisson process. An application to skew standard Young tableaux is given.

Article information

Electron. J. Probab., Volume 10 (2005), paper no. 40, 1359-1380.

Accepted: 18 November 2005
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J65: Brownian motion [See also 58J65]

continuous time random walk Brownian motion collision time skew Young tableaux tandem queue

This work is licensed under aCreative Commons Attribution 3.0 License.


Puchala, Zbigniew; Rolski, Tomasz. The Exact Asymptotic of the Time to Collision. Electron. J. Probab. 10 (2005), paper no. 40, 1359--1380. doi:10.1214/EJP.v10-291. https://projecteuclid.org/euclid.ejp/1464816841

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