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 Electronic Communications in Probability > Vol. 7 (2002) > Paper 1 open journal systems 


A Representation for Non-Colliding Random Walks

Neil O'Connell, BRIMS, HP Labs
Marc Yor, Universite Pierre et Marie Curie


Abstract
We define a sequence of mappings $Gamma_k:D_0(R_+)^kto D_0(R_+)^k$ and prove the following result: Let $N_1,ldots,N_n$ be the counting functions of independent Poisson processes on $R_+$ with respective intensities $mu_1<mu_2<cdots <mu_n$. The conditional law of $N_1,ldots,N_n$, given that $$N_1(t)lecdotsle N_n(t), mbox{ for all }tge 0,$$ is the same as the unconditional law of $Gamma_n(N)$. From this, we deduce the corresponding results for independent Poisson processes of equal rates and for independent Brownian motions (in both of these cases the conditioning is in the sense of Doob).  This extends a recent observation, independently due to Baryshnikov (2001) and Gravner, Tracy and Widom (2001), which relates the law of a certain functional of Brownian motion to that  of the largest eigenvalue of a GUE random matrix.  Our main result can also be regarded as a generalisation of Pitman's representation for the 3-dimensional Bessel process.


Full text: PDF

Pages: 1-12

Published on: July 28, 2001
Bibliography
  1. J. Baik (2000), Random vicious walks and random matrices. Comm. Pure Appl. Math. 53, 1385-1410. Math. Review 1 773 413
  2. J. Baik, P. Deift and K. Johansson (1999), On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12, no. 4, 1119-1178. Math. Review 2000e:05006
  3. Yu. Baryshnikov (2001), GUES and QUEUES. Probab. Theor. Rel. Fields 119, 256-274. Math. Review 1 818 248
  4. Ph. Biane (1994), Quelques propriétés du mouvement brownien dans un cone.. Stoch. Proc. Appl. 53, no. 2, 233-240. Math. Review 95j:60129
  5. Ph. Bougerol and Th. Jeulin (2001), Paths in Weyl chambers and random matrices. In preparation. Math. Review number not available.
  6. P. Brémaud (1981), Point Processes and Queues: Martingale Dynamics. Springer-Verlag. Math. Review 82m:60058
  7. P. Brémaud (1999), Markov Chains. Gibbs Fields, Monte-Carlo Simulation, and Queues. Texts in App. Maths., vol. 31. Springer. Math. Review 2000k:60137
  8. P.J. Burke (1956), The output of a queueing system. Operations Research 4, no. 6, 699--704. Math. Review 18,707g
  9. E. Cépa and D. Lépingle (1997), Diffusing particles with electrostatic repulsion. Probab. Th. Rel. Fields 107, no. 4, 429-449. Math. Review 98k:60177
  10. J.L. Doob (1984), Classical Potential Theory and its Probabilistic Counterpart. Springer. Math. Review 1 814 344
  11. F.J Dyson (1962), A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys. 3, 1191-1198. Math. Review 26 #5904
  12. S.N. Ethier and T.G. Kurtz (1986), Markov Processes: Characterization and Convergence. Wiley, New York. Math. Review 88a:60130
  13. P.J. Forrester (1999). Random walks and random permutations. Preprint, 1999. (xxx math.CO/9907037) Math. Review number not available.
  14. P.W. Glynn and W. Whitt (1991), Departures from many queues in series. Ann. Appl. Prob. 1, no. 4, 546-572. Math. Review 92i:60162
  15. D. Grabiner (1999), Brownian motion in a Weyl chamber, non-colliding particles, and random matrices. Ann. Inst. H. Poincaré Probab. Statist. 35, no. 2, 177-204. Math. Review 2000i:60091
  16. J. Gravner, C.A. Tracy and H. Widom (2001), Limit theorems for height fluctuations in a class of discrete space and time growth models. J. Stat. Phys. 102, nos. 5-6, 1085-1132. Math. Review number not available.
  17. J. M. Harrison and R.J. Williams (1990), On the quasireversibility of a multiclass Brownian service station. Ann. Probab. 18, 1249-1268. Math. Review 91i:60204
  18. D. Hobson and W. Werner (1996), Non-colliding Brownian motion on the circle. Bull. Math. Soc. 28, 643-650. Math. Review 97k:60217
  19. K. Johansson (2000), Shape fluctuations and random matrices. Commun. Math. Phys. 209, 437-476. Math. Review 1 737 991
  20. K. Johansson (1999), Discrete orthogonal polynomial ensembles and the Plancherel measure. Preprint 1999, to appear in Ann. Math. (xxx math.CO/9906120) Math. Review number not available.
  21. F.P. Kelly (1979), Reversibility and Stochastic Networks. Wiley. Math. Review 81j:60105
  22. Wolfgang König and Neil O'Connell (2001), Eigenvalues of the Laguerre process as non-colliding squared Bessel processes. To appear in Elect. Commun. Probab.. Math. Review number not available.
  23. Wolfgang König, Neil O'Connell and Sebastien Roch (2001), Non-colliding random walks, tandem queues and discrete ensembles. Preprint. Math. Review number not available.
  24. P.M. Morse (1955), Stochastic properties of waiting lines. Operations Research 3, 256. Math. Review 17,51d
  25. H. Matsumoto and M. Yor (1999), A version of Pitman's 2M-X theorem for geometric Brownian motions. C.R. Acad. Sci. Paris 328, Série I, 1067-1074. Math. Review 2000d:60134
  26. M.L. Mehta (1991), Random Matrices: Second Edition. Academic Press. Math. Review 92f:82002
  27. G.G. O'Brien (1954), The solution of some queueing problems. J. Soc. Indust. Appl. Math. 2, 134. Math. Review 16,600h
  28. Neil O'Connell and Marc Yor (2001), Brownian analogues of Burke's theorem. Stoch. Proc. Appl., to appear. Math. Review number not available.
  29. J. W. Pitman (1975), One-dimensional Brownian motion and the three-dimensional Bessel process. Adv. Appl. Probab. 7, 511-526. Math. Review 51 #11677
  30. J. W. Pitman and L.C.G. Rogers (1981), Markov functions. Ann. Probab. 9, 573-582. Math. Review 82j:60133
  31. E. Reich (1957), Waiting times when queues are in tandem. Ann. Math. Statist. 28, 768-773. Math. Review 19,1203b
  32. Ph. Robert (2000), Réseaux et files d'attente: méthodes probabilistes. Math. et Applications, vol. 35. Springer. Math. Review number not available.
  33. C.A. Tracy and H. Widom (1994), Fredholm determinants, differential equations and matrix models. Comm. Math. Phys. 163, no. 1, 33-72. Math. Review 95e:82005
  34. David Williams (1979), Diffusions, Markov Processes and Martingales. Volume 1: Foundations. Wiley. Math. Review 80i:60100
  35. David Williams (1974), Path decomposition and continuity of local time for one-dimensional diffusions I. Proc. London Math. Soc. 28, no. 3, 738-768. Math. Review 50 #3373













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Electronic Communications in Probability. ISSN: 1083-589X