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.


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