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2014 Excursions of excited random walks on integers
Elena Kosygina, Martin Zerner
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Electron. J. Probab. 19: 1-25 (2014). DOI: 10.1214/EJP.v19-2940


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.


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Elena Kosygina. Martin Zerner. "Excursions of excited random walks on integers." Electron. J. Probab. 19 1 - 25, 2014.


Accepted: 28 February 2014; Published: 2014
First available in Project Euclid: 4 June 2016

zbMATH: 1291.60093
MathSciNet: MR3174837
Digital Object Identifier: 10.1214/EJP.v19-2940

Primary: 60G50
Secondary: 60F17 , 60J70 , 60J80 , 60J85 , 60K37

Keywords: branching process , Cookie walk , diffusion approximation , excited random walk , excursion , Return time , squared Bessel process , strong transience

Vol.19 • 2014
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