## Electronic Journal of Probability

### Conditional Limit Theorems for Ordered Random Walks

#### Abstract

In a recent paper of Eichelsbacher and Koenig (2008) the model of ordered random walks has been considered. There it has been shown that, under certain moment conditions, one can construct a $k$-dimensional random walk conditioned to stay in a strict order at all times. Moreover, they have shown that the rescaled random walk converges to the Dyson Brownian motion. In the present paper we find the optimal moment assumptions for the construction proposed by Eichelsbacher and Koenig, and generalise the limit theorem for this conditional process.

#### Article information

Source
Electron. J. Probab. Volume 15 (2010), paper no. 11, 292-322.

Dates
Accepted: 8 April 2010
First available in Project Euclid: 1 June 2016

https://projecteuclid.org/euclid.ejp/1464819796

Digital Object Identifier
doi:10.1214/EJP.v15-752

Mathematical Reviews number (MathSciNet)
MR2609589

Zentralblatt MATH identifier
1201.60040

Rights

#### Citation

Denisov, Denis; Wachtel, Vitali. Conditional Limit Theorems for Ordered Random Walks. Electron. J. Probab. 15 (2010), paper no. 11, 292--322. doi:10.1214/EJP.v15-752. https://projecteuclid.org/euclid.ejp/1464819796

#### References

• Baik, J. and Suidan, T. Random matrix central limit theorems for nonintersecting random walks. Ann. Probab. 35 (2007), 1807–1834.
• Bertoin, J. and Doney, R.A. On conditioning a random walk to stay nonnegative. Ann. Probab. 22 (1994), 2152–2167.
• Bodineau, T. and Martin, J. A universality property for last-passage percolation paths close to the axis. Electron. Comm. in Probab. 10 (2005), 105–112.
• Borisov, I.S. On the question of the rate of convergence in the Donsker–Prokhorov invariance principle. Theory Probab. Appl. 28 (1983),388–392.
• Borovkov, A.A. Notes on inequalities for sums of independent random variables. Theory Probab. Appl. 17 (1972), 556–557.
• Bryn-Jones, A. and Doney, R.A. A functional limit theorem for random walk conditioned to stay non-negative. J. London Math. Soc. (2) 74 2006, 244–258.
• Dyson F.J. A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys 3 (1962), 1191–1198.
• Eichelsbacher, P. and König, W. Ordered random walks. Electron. J. Probab. 13, (2008) 1307–1336.
• Grabiner, D.J. Brownian motion in a Weyl chamber, non-colliding particles, and random matrices. Ann. Inst. H. Poincare Probab. Statist., 35(2) (1999), 177–204.
• König, W., O'Connell, N. and Roch, S. Non-colliding random walks, tandem queues, and discrete orthogonal polynomial ensembles. Electron. J. Probab. 7, (2002), 1–24.
• König, W. Orthogonal polynomial ensembles in probability theory. Probab. Surv. 2 (2005), 385–447.
• König, W and Schmid, P. Random walks conditioned to stay in Weyl chambers of type C and D. arXiv:0911.0631 (2009).
• Major, P. The approximation of partial sums of rv's. Z. Wahrscheinlichkeitstheorie verw. Gebiete 35 (1976), 213–220.
• Nagaev, S.V. Large deviations of sums of independent random variables. Ann. Probab. 7 (1979), 745–789.
• O'Connell, N. and Yor, M. A representation for non-colliding random walks. Elect. Comm. Probab. 7 (2002), 1–12.
• Puchala, Z, Rolski, T. The exact asymptotic of the collision time tail distribution for independent Brownian particles with different drifts. Probab. Theory Related Fields 142 (2008), 595–617.
• Schapira, B. Random walk on a building of type $\tilde{A}_r$ and Brownian motion on a Weyl chamber. Ann. Inst. H. Poincare Probab. Statist. 45 (2009), 289-301.
• Varopoulos, N.Th. Potential theory in conical domains. Math. Proc. Camb. Phil. Soc. 125 (1999), 335–384.