## Illinois Journal of Mathematics

### Random walks with occasionally modified transition probabilities

#### Abstract

We study recurrence properties and the validity of the (weak) law of large numbers for (discrete time) processes which, in the simplest case, are obtained from simple symmetric random walk on $\mathbb{Z}$ by modifying the distribution of a step from a fresh point. If the process is denoted as $\{S_n\}_{n \ge0}$, then the conditional distribution of $S_{n+1} - S_n$ given the past through time $n$ is the distribution of a simple random walk step, provided $S_n$ is at a point which has been visited already at least once during $[0,n-1]$. Thus, in this case, $P\{S_{n+1}-S_n = \pm1|S_\ell, \ell\le n\} = 1/2$. We denote this distribution by $P_1$. However, if $S_n$ is at a point which has not been visited before time $n$, then we take for the conditional distribution of $S_{n+1}-S_n$, given the past, some other distribution $P_2$. We want to decide in specific cases whether $S_n$ returns infinitely often to the origin and whether $(1/n)S_n \to0$ in probability. Generalizations or variants of the $P_i$ and the rules for switching between the $P_i$ are also considered.

#### Article information

Source
Illinois J. Math., Volume 54, Number 4 (2010), 1213-1238.

Dates
First available in Project Euclid: 24 September 2012

https://projecteuclid.org/euclid.ijm/1348505527

Digital Object Identifier
doi:10.1215/ijm/1348505527

Mathematical Reviews number (MathSciNet)
MR2981846

Zentralblatt MATH identifier
1259.60110

#### Citation

Raimond, Olivier; Schapira, Bruno. Random walks with occasionally modified transition probabilities. Illinois J. Math. 54 (2010), no. 4, 1213--1238. doi:10.1215/ijm/1348505527. https://projecteuclid.org/euclid.ijm/1348505527

#### References

• I. Benjamini and D. B. Wilson, Excited random walk, Electron. Comm. Probab. 8 (2003), 86–92 (electronic).
• P. Billingsley, Probability and measure, 2nd ed., John Wiley & Sons, New York, 1986.
• R. Durett, H. Kesten and G. Lawler, Making money from fair games, Random walks, Brownian motion, and interacting particle systems, Progr. Probab., vol. 28, Birkhäuser Boston, Boston, MA, 1991, pp. 255–267.
• D. Dolgopyat, Central limit theorem for excited random walk in the recurrent regime, ALEA 8 (2011), 259–268.
• F. Merkl and S. W. W. Rolles, Recurrence of edge reinforced random walk on a two-dimensional graph, Ann. Probab. 37 (2009), 1679–1714.
• W. Feller, An introduction to probability theory and its applications, vol.II, 2nd ed., John Wiley & Sons, New York, 1971.
• N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, 2nd ed., North-Holland Mathematical Library, Amsterdam; Kodansha, Ltd., Tokyo, 1989, xvi+555 pp.
• J. Kent, Some probabilistic properties of Bessel functions, Ann. Probab. 6 (1978), 760–770.
• H. Kesten, Recurrence criteria for multi-dimensional Markov chains and multi-dimensional linear birth and death processes, Adv. in Appl. Probab. 8 (1976), 58–87.
• H. Kesten and G. Lawler, A necessary condition for making money from fair games, Ann. Probab. 20 (1992), 855–882.
• E. Kosygina and T. Mountford, Limit laws of transient excited random walks on integers, to appear in Stochast. Process. Appl.
• N. N. Lebedev, Special functions and their applications, revised ed., Dover Publications, Inc., New York, 1972, translated from the Russian and edited by Richard A. Silverman.
• D. Revuz and M. Yor, Continuous martingales and Brownian motion, 3rd ed., Springer-Verlag, Berlin, 1999.
• D. Williams, Probability with martingales, Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge, 1991.