## 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) |
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Pages (from-to) | 162-173 |

Number of pages | 12 |

Journal | Journal of Applied Probability |

Volume | 51 |

Issue number | 1 |

DOIs | |

State | Published - Mar 2014 |

## Keywords

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

## ASJC Scopus subject areas

- Statistics and Probability
- General Mathematics
- Statistics, Probability and Uncertainty