Scaling limits of recurrent excited random walks on integers
Elena Kosygina (Baruch College and the CUNY Graduate Center)
Abstract
We describe scaling limits of recurrent excited random walks (ERWs) on $\mathbb{Z}$ in i.i.d. cookie environments with a bounded number of cookies per site. We allow both positive and negative excitations. It is known that ERW is recurrent if and only if the expected total drift per site, $\delta$, belongs to the interval $[-1,1]$. We show that if $|\delta|<1$ then the diffusively scaled ERW under the averaged measure converges to a $(\delta,-\delta)$-perturbed Brownian motion. In the boundary case, $|\delta|=1$, the space scaling has to be adjusted by an extra logarithmic term, and the weak limit of ERW happens to be a constant multiple of the running maximum of the standard Brownian motion, a transient process.
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Pages: 1-14
Publication Date: August 9, 2012
DOI: 10.1214/ECP.v17-2213
References
- Aldous, David. Stopping times and tightness. II. Ann. Probab. 17 (1989), no. 2, 586--595. MR0985380
- Basdevant, Anne-Laure; Singh, Arvind. On the speed of a cookie random walk. Probab. Theory Related Fields 141 (2008), no. 3-4, 625--645. MR2391167
- Basdevant, Anne-Laure; Singh, Arvind. Rate of growth of a transient cookie random walk. Electron. J. Probab. 13 (2008), no. 26, 811--851. MR2399297
- Benjamini, Itai; Wilson, David B. Excited random walk. Electron. Comm. Probab. 8 (2003), 86--92 (electronic). MR1987097
- Bérard, Jean; Ramírez, Alejandro. Central limit theorem for the excited random walk in dimension $D\geq 2$. Electron. Comm. Probab. 12 (2007), 303--314 (electronic). MR2342709
- Billingsley, Patrick. Convergence of probability measures. Second edition. Wiley Series in Probability and Statistics: Probability and Statistics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999. x+277 pp. ISBN: 0-471-19745-9 MR1700749
- Chaumont, L.; Doney, R. A. Pathwise uniqueness for perturbed versions of Brownian motion and reflected Brownian motion. Probab. Theory Related Fields 113 (1999), no. 4, 519--534. MR1717529
- Davis, Burgess. Weak limits of perturbed random walks and the equation $Y_ t=B_ t+\alpha\sup\{Y_ s\colon\ s\leq t\}+\beta\inf\{Y_ s\colon\ s\leq t\}$. Ann. Probab. 24 (1996), no. 4, 2007--2023. MR1415238
- Dolgopyat, Dmitry. Central limit theorem for excited random walk in the recurrent regime. ALEA Lat. Am. J. Probab. Math. Stat. 8 (2011), 259--268. MR2831235
- Durrett, Rick. Probability: theory and examples. Fourth edition. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2010. x+428 pp. ISBN: 978-0-521-76539-8 MR2722836
- Gikhman, I. I.; Skorokhod, A. V. Introduction to the theory of random processes. Translated from the Russian by Scripta Technica, Inc. W. B. Saunders Co., Philadelphia, Pa.-London-Toronto, Ont. 1969 xiii+516 pp. MR0247660
- Gīhman, Ĭ. Ī.; Skorohod, A. V. The theory of stochastic processes. I. Translated from the Russian by S. Kotz. Die Grundlehren der mathematischen Wissenschaften, Band 210. Springer-Verlag, New York-Heidelberg, 1974. viii+570 pp. MR0346882
- Hall, P.; Heyde, C. C. Martingale limit theory and its application. Probability and Mathematical Statistics. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. xii+308 pp. ISBN: 0-12-319350-8 MR0624435
- Kosygina, Elena; Zerner, Martin P. W. Positively and negatively excited random walks on integers, with branching processes. Electron. J. Probab. 13 (2008), no. 64, 1952--1979. MR2453552
- Kosygina, Elena; Mountford, Thomas. Limit laws of transient excited random walks on integers. Ann. Inst. Henri Poincaré Probab. Stat. 47 (2011), no. 2, 575--600. MR2814424
- M. Menshikov, S. Popov, A. Ramirez, M. Vachkovskaya. On a general many-dimensional excited random walk. Ann. Probab. To appear.
- Mountford, Thomas; Pimentel, Leandro P. R.; Valle, Glauco. On the speed of the one-dimensional excited random walk in the transient regime. ALEA Lat. Am. J. Probab. Math. Stat. 2 (2006), 279--296. MR2285733
- Perman, Mihael; Werner, Wendelin. Perturbed Brownian motions. Probab. Theory Related Fields 108 (1997), no. 3, 357--383. MR1465164
- J. Peterson. Large deviations and slowdown asymptotics for one-dimensional excited random walks. arXiv:1201.0318.
- R. Rastegar, A. Roiterstein. Maximum occupation time of a transient excited random walk on ${\bf Z}$. arXiv:1111.1254.
- Tóth, Bálint. Generalized Ray-Knight theory and limit theorems for self-interacting random walks on ${\bf Z}^ 1$. Ann. Probab. 24 (1996), no. 3, 1324--1367. MR1411497
- Zerner, Martin P. W. Multi-excited random walks on integers. Probab. Theory Related Fields 133 (2005), no. 1, 98--122. MR2197139
- Zerner, Martin P. W. Recurrence and transience of excited random walks on $\Bbb Z^ d$ and strips. Electron. Comm. Probab. 11 (2006), 118--128 (electronic). MR2231739
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