## Advances in Applied Probability

- Adv. in Appl. Probab.
- Volume 48, Number 1 (2016), 215-234.

### The speed of a random walk excited by its recent history

#### Abstract

Let *N* and *M* be positive integers satisfying 1≤ *M*≤ *N*, and let 0< *p*_{0} < *p*_{1} < 1. Define a process {X_{n}}_{n=0}^{∞} on 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*→ ∞ and *M*/*N*→ *r*∈[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.

#### Article information

**Source**

Adv. in Appl. Probab., Volume 48, Number 1 (2016), 215-234.

**Dates**

First available in Project Euclid: 8 March 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.aap/1457466163

**Mathematical Reviews number (MathSciNet)**

MR3473575

**Zentralblatt MATH identifier**

1337.60085

**Subjects**

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60F15: Strong theorems

**Keywords**

Random walk with internal states excited random walk

#### Citation

Pinsky, Ross G. The speed of a random walk excited by its recent history. Adv. in Appl. Probab. 48 (2016), no. 1, 215--234. https://projecteuclid.org/euclid.aap/1457466163