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The exact asymptotic of the time to collision

Wed, 2008-03-05 21:57 — Zbyszek

Authors:

Puchała, Z.; Rolski, T.

Source:

Electronic Journal of Probability, Volume 10, Issue Probab, Number 40, p.1359-1380 (2005)

URL:

http://www.math.washington.edu/~ejpecp/viewarticle.php?id=1551&layout=abstract

Keywords:

continuous time random walk; Brownian motion; Collision time; skew Young tableaux; tandem queue; Probability Theory; Stochastic Processes

Abstract:

Abstract 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.