## Journal of Applied Probability

- J. Appl. Probab.
- Volume 51, Number 1 (2014), 162-173.

### The maximum of a symmetric next neighbor walk on the nonnegative integers

Ora E. Percus and Jerome K. Percus

#### Abstract

We consider a one-dimensional discrete symmetric random walk with a reflecting boundary at the origin.
Generating functions are found for the two-dimensional probability distribution
P{*S*_{n} = *x*, max_{1≤j≤n}*S*_{n} = *a*}
of being at position *x* after *n* steps, while the maximal location that the walker has achieved
during these *n* steps is *a*. We also obtain the familiar (marginal) one-dimensional distribution for
*S*_{n} = *x*, but more importantly that for
max_{1≤j≤n}*S*_{j} = *a* asymptotically at fixed
*a*^{2} / *n*. We are able to compute and compare the expectations and variances of the two
one-dimensional distributions, finding that they have qualitatively similar forms, but differ quantitatively in
the anticipated fashion.

#### Article information

**Source**

J. Appl. Probab., Volume 51, Number 1 (2014), 162-173.

**Dates**

First available in Project Euclid: 25 March 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.jap/1395771421

**Digital Object Identifier**

doi:10.1239/jap/1395771421

**Mathematical Reviews number (MathSciNet)**

MR3189449

**Zentralblatt MATH identifier**

1306.60046

**Subjects**

Primary: 60G50: Sums of independent random variables; random walks

Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

**Keywords**

One-dimensional random walk statistics of the maximum discrete probability asymptotic techniques

#### Citation

Percus, Ora E.; Percus, Jerome K. The maximum of a symmetric next neighbor walk on the nonnegative integers. J. Appl. Probab. 51 (2014), no. 1, 162--173. doi:10.1239/jap/1395771421. https://projecteuclid.org/euclid.jap/1395771421