### 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 ¥[Sn = x, maxi<7< Sn = 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 Sn - x, but more importantly that for maxi<7< Sj = a asymptotically at fixed a2/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.

Original language | English (US) |
---|---|

Pages (from-to) | 162-173 |

Number of pages | 12 |

Journal | Journal of Applied Probability |

Volume | 51 |

Issue number | 1 |

DOIs | |

State | Published - 2014 |

### Fingerprint

### Keywords

- Asymptotic techniques
- Discrete probability
- One-dimensional random walk
- Statistics of the maximum

### ASJC Scopus subject areas

- Mathematics(all)
- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Journal of Applied Probability*,

*51*(1), 162-173. https://doi.org/10.1239/jap/1395771421

**The maximum of a symmetric next neighbor walk on the nonnegative integers.** / Percus, Ora E.; Percus, Jerome.

Research output: Contribution to journal › Article

*Journal of Applied Probability*, vol. 51, no. 1, pp. 162-173. https://doi.org/10.1239/jap/1395771421

}

TY - JOUR

T1 - The maximum of a symmetric next neighbor walk on the nonnegative integers

AU - Percus, Ora E.

AU - Percus, Jerome

PY - 2014

Y1 - 2014

N2 - 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 ¥[Sn = x, maxi<7< Sn = 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 Sn - x, but more importantly that for maxi<7< Sj = a asymptotically at fixed a2/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.

AB - 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 ¥[Sn = x, maxi<7< Sn = 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 Sn - x, but more importantly that for maxi<7< Sj = a asymptotically at fixed a2/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.

KW - Asymptotic techniques

KW - Discrete probability

KW - One-dimensional random walk

KW - Statistics of the maximum

UR - http://www.scopus.com/inward/record.url?scp=84903767071&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84903767071&partnerID=8YFLogxK

U2 - 10.1239/jap/1395771421

DO - 10.1239/jap/1395771421

M3 - Article

AN - SCOPUS:84903767071

VL - 51

SP - 162

EP - 173

JO - Journal of Applied Probability

JF - Journal of Applied Probability

SN - 0021-9002

IS - 1

ER -