Electronic Journal of Probability

The Exact Asymptotic of the Time to Collision

Zbigniew Puchala and Tomasz Rolski

Full-text: Open access

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.

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464816841

Digital Object Identifier
doi:10.1214/EJP.v10-291

Mathematical Reviews number (MathSciNet)
MR2183005

Zentralblatt MATH identifier
1110.60069

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

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

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

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


Export citation

References

  • M. Abramowitz, I.A. Stegun, I.A. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards (1964).
  • S. Asmussen, Applied Probability and Queues. Second Ed.,Springer (2003), New York.
  • A. Dembo, O. Zeitouni Large Deviations Techniques and Applications. Jones and Bartlett (1993), Boston.
  • Y. Doumerc, N. O'Connell, Exit problems associated with finite reflection groups. Probability Theory and Related Fields 132 (2005), 501 - 538.
  • W. Fulton Young Tableaux. Cambridge University Press (1997), Cambridge.
  • D.J. Grabiner, Brownian motion in a Weyl chamber, non-colliding particles, and random matrices. Ann. Inst. H. Poincaré. Probab. Statist. bf 35 (1999), 177-204.
  • K. Knopp, Theorie und Anwendung der unendlichen Reichen 4th Ed., Springer-Verlag (1947), Berlin and Heidelberg.
  • I.G. Macdonald, Symetric Functions and Hall Polynomials.Clarendon Press (1979), Oxford.
  • W. Massey (1987) Calculating exit times for series Jackson networks. J. Appl. Probab., 24/1.
  • M.L. Mehta, Random Matrices. Second edition. Academic Press (1991), Boston.
  • Z. Puchala, A proof of Grabiner's theorem on non-colliding particles. Probability and Mathematical Statistics (2005).
  • A. Regev, Asymptotic values for degrees associated with stripes of Young diagrams, Adv. Math. 41 (1981), 115-136.
  • G.N. Watson, A Treatise on the Theory of Bessel Functions 2nd ed. Cambridge University Press, (1944), Cambridge.
  • E.W. Weisstein, Power Sum. From MathWorld – A Wolfram Web Resource. http://mathworld.wolfram.com/PowerSum.html.