Electronic Journal of Probability
- Electron. J. Probab.
- Volume 19 (2014), paper no. 25, 25 pp.
Excursions of excited random walks on integers
Several phase transitions for excited random walks on the integers are known to be characterized by a certain drift parameter $\delta\in\mathbb R$. For recurrence/transience the critical threshold is $|\delta|=1$, for ballisticity it is $|\delta|=2$ and for diffusivity $|\delta|=4$. In this paper we establish a phase transition at $|\delta|=3$. We show that the expected return time of the walker to the starting point, conditioned on return, is finite iff $|\delta|>3$. This result follows from an explicit description of the tail behaviour of the return time as a function of $\delta$, which is achieved by diffusion approximation of related branching processes by squared Bessel processes.
Electron. J. Probab., Volume 19 (2014), paper no. 25, 25 pp.
Accepted: 28 February 2014
First available in Project Euclid: 4 June 2016
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60K37: Processes in random environments 60F17: Functional limit theorems; invariance principles 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J85: Applications of branching processes [See also 92Dxx]
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Kosygina, Elena; Zerner, Martin. Excursions of excited random walks on integers. Electron. J. Probab. 19 (2014), paper no. 25, 25 pp. doi:10.1214/EJP.v19-2940. https://projecteuclid.org/euclid.ejp/1465065667