## Electronic Journal of Probability

### Ordered random walks with heavy tails

#### Abstract

This note continues paper of Denisov and Wachtel (2010), where we have constructed a $k$-dimensional random walk conditioned to stay in the Weyl chamber of type $A$. The  construction was done  under the assumption that the original random walk has $k-1$ moments. In this note we continue the study of killed random walks in the Weyl chamber, and assume that the tail of increments is regularly varying of index $\alpha<k-1$. It appears that the asymptotic behaviour of random walks is different in this case. We determine the asymptotic behaviour of the exit time, and, using this information, construct a conditioned process which lives on a partial compactification of the Weyl chamber.

#### Article information

Source
Electron. J. Probab., Volume 17 (2012), paper no. 4, 21 pp.

Dates
Accepted: 11 January 2012
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465062326

Digital Object Identifier
doi:10.1214/EJP.v17-1719

Mathematical Reviews number (MathSciNet)
MR2878783

Zentralblatt MATH identifier
1246.60069

Subjects
Primary: 60G50: Sums of independent random variables; random walks

Rights

#### Citation

Denisov, Denis; Wachtel, Vitali. Ordered random walks with heavy tails. Electron. J. Probab. 17 (2012), paper no. 4, 21 pp. doi:10.1214/EJP.v17-1719. https://projecteuclid.org/euclid.ejp/1465062326

#### References

• Bertoin, J.; Doney, R. A. On conditioning a random walk to stay nonnegative. Ann. Probab. 22 (1994), no. 4, 2152–2167.
• Denisov, Denis; Shneer, Vsevolod. Asymptotics for first-passage times of Lévy processes and random walks. ArXiv Preprint 0712.0728.
• Denisov, D.; Dieker, A. B.; Shneer, V. Large deviations for random walks under subexponentiality: the big-jump domain. Ann. Probab. 36 (2008), no. 5, 1946–1991.
• Denisov, Denis; Wachtel, Vitali. Conditional limit theorems for ordered random walks. Electron. J. Probab. 15 (2010), no. 11, 292–322.
• Doney, R. A. The Martin boundary and ratio limit theorems for killed random walks. J. London Math. Soc. (2) 58 (1998), no. 3, 761–768.
• Eichelsbacher, Peter; König, Wolfgang. Ordered random walks. Electron. J. Probab. 13 (2008), no. 46, 1307–1336.
• Ignatiouk-Robert, Irina. Martin boundary of a killed random walk on a half-space. J. Theoret. Probab. 21 (2008), no. 1, 35–68.
• Ignatiouk-Robert, Irina; Loree, Christophe. Martin boundary of a killed random walk on a quadrant. Ann. Probab. 38 (2010), no. 3, 1106–1142.
• Ignatiouk-Robert, I. Martin boundary of a killed random walk on mathbbZ_+^d. ArXiv Preprint 0909.3921.
• Jacka, Saul; Warren, Jon. Examples of convergence and non-convergence of Markov chains conditioned not to die. Electron. J. Probab. 7 (2002), no. 1, 22 pp. (electronic).
• König, Wolfgang; Schmid, Patrick. Random walks conditioned to stay in Weyl chambers of type C and D. Electron. Commun. Probab. 15 (2010), 286–296.
• Nagaev, S. V. Large deviations of sums of independent random variables. Ann. Probab. 7 (1979), no. 5, 745–789.
• Raschel, Kilian. Green functions and Martin compactification for killed random walks related to $\rm SU(3)$. Electron. Commun. Probab. 15 (2010), 176–190.
• Raschel, K. Green functions for killed random walk in the Weyl chamber of Sp(4). ph Ann. Inst. H. Poincaré Probab. Statist., 47, (2011), 1001–1019.
• Rosenthal, Haskell P. On the subspaces of $L^{p}$ $(p>2)$ spanned by sequences of independent random variables. Israel J. Math. 8 1970 273–303.