Electronic Journal of Probability

Ordered Random Walks

Peter Eichelsbacher and Wolfgang König

Full-text: Open access

Abstract

We construct the conditional version of $k$ independent and identically distributed random walks on $R$ given that they stay in strict order at all times. This is a generalisation of so-called non-colliding or non-intersecting random walks, the discrete variant of Dyson's Brownian motions, which have been considered yet only for nearest-neighbor walks on the lattice. Our only assumptions are moment conditions on the steps and the validity of the local central limit theorem. The conditional process is constructed as a Doob $h$-transform with some positive regular function $V$ that is strongly related with the Vandermonde determinant and reduces to that function for simple random walk. Furthermore, we prove an invariance principle, i.e., a functional limit theorem towards Dyson's Brownian motions, the continuous analogue.

Article information

Source
Electron. J. Probab., Volume 13 (2008), paper no. 46, 1307-1336.

Dates
Accepted: 14 August 2008
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819119

Digital Object Identifier
doi:10.1214/EJP.v13-539

Mathematical Reviews number (MathSciNet)
MR2430709

Zentralblatt MATH identifier
1189.60092

Subjects
Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60F17: Functional limit theorems; invariance principles

Keywords
Dyson's Brownian motions Vandermonde determinant Doob h-transform non-colliding random walks non-intersecting random processes fluctuation theory

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

Citation

Eichelsbacher, Peter; König, Wolfgang. Ordered Random Walks. Electron. J. Probab. 13 (2008), paper no. 46, 1307--1336. doi:10.1214/EJP.v13-539. https://projecteuclid.org/euclid.ejp/1464819119


Export citation

References

  • Baik, Jinho. Random vicious walks and random matrices. Comm. Pure Appl. Math. 53 (2000), no. 11, 1385–1410.
  • Baik, Jinho; Suidan, Toufic M. Random matrix central limit theorems for nonintersecting random walks. Ann. Probab. 35 (2007), no. 5, 1807–1834.
  • Bertoin, J.; Doney, R. A. On conditioning a random walk to stay nonnegative. Ann. Probab. 22 (1994), no. 4, 2152–2167.
  • Bougerol, Philippe; Jeulin, Thierry. Paths in Weyl chambers and random matrices. Probab. Theory Related Fields 124 (2002), no. 4, 517–543.
  • Bru, Marie-France. Wishart processes. J. Theoret. Probab. 4 (1991), no. 4, 725–751.
  • Deift, P. A. Orthogonal polynomials and random matrices: a Riemann-Hilbert approach.Courant Lecture Notes in Mathematics, 3. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999. viii+273 pp. ISBN: 0-9658703-2-4; 0-8218-2695-6
  • Y. Doumerc, Matrices aleatoires, processus stochastiques et groupes de reflexions, PhD thesis, Universite Toulouse (2005).
  • F.J. Dyson, A Brownian-motion model for the eigenvalues of a random matrix, J.Math.Phys. 3, 1191–1198 (1962).
  • Eichelsbacher, Peter; Löwe, Matthias. Moderate deviations for i.i.d. random variables. ESAIM Probab. Stat. 7 (2003), 209–218 (electronic).
  • Feller, William. An introduction to probability theory and its applications. Vol. II.Second edition John Wiley & Sons, Inc., New York-London-Sydney 1971 xxiv+669 pp.
  • D. Grabiner, Brownian motion in a Weyl chamber, non-colliding particles, and random matrices, Ann. Inst. H. Poincare Probab.Statist. 35:2, 177–204 (1999).
  • Hiai, Fumio; Petz, Dénes. The semicircle law, free random variables and entropy.Mathematical Surveys and Monographs, 77. American Mathematical Society, Providence, RI, 2000. x+376 pp. ISBN: 0-8218-2081-8
  • Hobson, David G.; Werner, Wendelin. Non-colliding Brownian motions on the circle. Bull. London Math. Soc. 28 (1996), no. 6, 643–650.
  • Johansson, Kurt. Discrete orthogonal polynomial ensembles and the Plancherel measure. Ann. of Math. (2) 153 (2001), no. 1, 259–296.
  • Johansson, Kurt. Shape fluctuations and random matrices. Comm. Math. Phys. 209 (2000), no. 2, 437–476.
  • Johansson, Kurt. Non-intersecting paths, random tilings and random matrices. Probab. Theory Related Fields 123 (2002), no. 2, 225–280.
  • Karlin, Samuel; McGregor, James. Coincidence probabilities. Pacific J. Math. 9 1959 1141–1164.
  • M. Katori and H. Tanemura, Scaling limit of vicious walks and two-matrix model, Phys.Rev.E. 66 011105 (2002).
  • Katori, Makoto; Tanemura, Hideki. Noncolliding Brownian motions and Harish-Chandra formula. Electron. Comm. Probab. 8 (2003), 112–121 (electronic).
  • Katori, Makoto; Tanemura, Hideki. Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems. J. Math. Phys. 45 (2004), no. 8, 3058–3085.
  • Katori, Makoto; Nagao, Taro; Tanemura, Hideki. Infinite systems of non-colliding Brownian particles. Stochastic analysis on large scale interacting systems, 283–306, Adv. Stud. Pure Math., 39, Math. Soc. Japan, Tokyo, 2004.
  • König, Wolfgang; O'Connell, Neil. Eigenvalues of the Laguerre process as non-colliding squared Bessel processes. Electron. Comm. Probab. 6 (2001), 107–114 (electronic).
  • König, Wolfgang; O'Connell, Neil; Roch, Sébastien. Non-colliding random walks, tandem queues, and discrete orthogonal polynomial ensembles. Electron. J. Probab. 7 (2002), no. 5, 24 pp. (electronic).
  • König, Wolfgang. Orthogonal polynomial ensembles in probability theory. Probab. Surv. 2 (2005), 385–447 (electronic).
  • O'Connell, Neil. Random matrices, non-colliding processes and queues. Séminaire de Probabilités, XXXVI, 165–182, Lecture Notes in Math., 1801, Springer, Berlin, 2003.
  • Petrov, V. V. Sums of independent random variables.Translated from the Russian by A. A. Brown.Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 82.Springer-Verlag, New York-Heidelberg, 1975. x+346 pp.