## The speed of a random walk excited by its recent history.(English)Zbl 1337.60085

Summary: Let $$N$$ and $$M$$ be positive integers satisfying $$1 \leq M \leq N$$, and let $$0 < p_{0} < p_{1} < 1$$. Define a process $$\{X_{n}\}_{n=0}^{\infty}$$ on $$\mathbb{Z}$$ as follows. At each step, the process jumps either one step to the right or one step to the left, according to the following mechanism. For the first $$N$$ steps, the process behaves like a random walk that jumps to the right with probability $$p_{0}$$ and to the left with probability $$1-p_{0}$$. At subsequent steps the jump mechanism is defined as follows: if at least $$M$$ out of the $$N$$ most recent jumps were to the right, then the probability of jumping to the right is $$p_{1}$$; however, if fewer than $$M$$ out of the $$N$$ most recent jumps were to the right then the probability of jumping to the right is $$p_{0}$$. We calculate the speed of the process. Then we let $$N \to \infty$$ and $$M/N \to r \in [0,1]$$, and calculate the limiting speed. More generally, we consider the above questions for a random walk with a finite number $$l$$ of threshold levels, $$(M_{i},p_{i})_{i=1}^{l}$$, above the pre-threshold level $$p_{0}$$, as well as for one model with $$l=N$$ such thresholds.

### MSC:

 60G50 Sums of independent random variables; random walks 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 60F15 Strong limit theorems

### Keywords:

excited random walk; internal states
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### References:

  Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications , 2nd edn. Springer, New York. · Zbl 0896.60013  Herrmann, S. and Vallois, P. (2010). From persistent random walk to the telegraph noise. Stoch. Dyn. 10 , 161-196. · Zbl 1196.60082  Hughes, B. (1995). Random Walks and Random Environments, Vol. 1 Random Walks . Oxford University Press. · Zbl 0820.60053  Kosygina, E. and Zerner, M. P. W. (2013). Excited random walks: results, methods, open problems. Bull. Inst. Math. Acad. Sinica (N.S.) 8 , 105-157. · Zbl 1304.60109  Pemantle, R. (2007). A survey of random processes with reinforcement. Prob. Surveys 4 , 1-79. · Zbl 1189.60138  Renshaw, E. and Henderson, R. (1981). The correlated random walk. J. Appl. Prob. 18 , 403-414. · Zbl 0479.60071
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