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 Electronic Journal of Probability > Vol. 10 (2005) > Paper 40 open journal systems 


The Exact Asymptotic of the Time to Collision

Zbigniew Puchala, Wroclaw University
Tomasz Rolski, Wroclaw University


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.


Full text: PDF

Pages: 1359-1380

Published on: November 18, 2005
Bibliography
  1. M. Abramowitz, I.A. Stegun, I.A. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards (1964). Math. Review 29 #4914
  2. S. Asmussen, Applied Probability and Queues. Second Ed.,Springer (2003), New York. Math. Review 2004f:60001
  3. A. Dembo, O. Zeitouni Large Deviations Techniques and Applications . Jones and Bartlett (1993), Boston. Math. Review 95a:60034
  4. Y. Doumerc, N. O'Connell, Exit problems associated with finite reflection groups. Probability Theory and Related Fields 132 (2005), 501 - 538 Math. Review number not available.
  5. W. Fulton Young Tableaux . Cambridge University Press (1997), Cambridge. Math. Review 99f:05119
  6. 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. Math. Review 2000i:60091
  7. K. Knopp, Theorie und Anwendung der unendlichen Reichen 4th Ed., Springer-Verlag (1947), Berlin and Heidelberg. Math. Review 10,446a
  8. I.G. Macdonald, Symetric Functions and Hall Polynomials .Clarendon Press (1979), Oxford. Math. Review 84g:05003
  9. W. Massey (1987) Calculating exit times for series Jackson networks. J. Appl. Probab. , 24/1. Math. Review 88d:60236
  10. M.L. Mehta, Random Matrices . Second edition. Academic Press (1991), Boston. Math. Review 92f:82002
  11. Z. Puchala, A proof of Grabiner's theorem on non-colliding particles. Probability and Mathematical Statistics (2005).
  12. A. Regev, Asymptotic values for degrees associated with stripes of Young diagrams, Adv. Math. 41 (1981), 115-136. Math. Review 82h:20015
  13. G.N. Watson, A Treatise on the Theory of Bessel Functions 2nd ed . Cambridge University Press, (1944), Cambridge. Math. Review 6,64a
  14. E.W. Weisstein, Power Sum . From MathWorld -- A Wolfram Web Resource. http://mathworld.wolfram.com/PowerSum.html. Math. Review number not available.
















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Electronic Journal of Probability. ISSN: 1083-6489