### Abstract

Our main goal is to study a class of processes whose increments are generated via a cellular automata rule. Given the increments of a simple biased random walk, a new sequence of (dependent) Bernoulli random variables is produced. It is built, from the original sequence, according to a cellular automata rule. Equipped with these two sequences, we construct two more according to the same cellular automata rule. The construction is repeated a fixed number of times yielding an infinite array ({−K,…,K}×N) of (dependent) Bernoulli random variables. Taking partial sums of these sequences, we obtain a (2K+1)-dimensional process whose increments belong to the state space {−1,1}^{2K+1}. The aim of the paper is to study the long term behaviour of this process. In particular, we establish transience/recurrence properties and prove an invariance principle. The limiting behaviour of these processes depends strongly on the direction of the iteration, and exhibits few surprising features. This work is motivated by an earlier investigation (see Collevecchio et al. (2015)), in which the starting sequence is symmetric, and by the related work Ferrari et al. (2000).

Original language | English |
---|---|

Pages (from-to) | 860-877 |

Number of pages | 18 |

Journal | Stochastic Processes and their Applications |

Volume | 129 |

Issue number | 3 |

DOIs | |

Publication status | Published - Mar 2019 |

### Keywords

- Bootstrap random walks
- Functional limit theorem
- Random walks

### Cite this

}

*Stochastic Processes and their Applications*, vol. 129, no. 3, pp. 860-877. https://doi.org/10.1016/j.spa.2018.03.022

**Invariance principle for biased bootstrap random walks.** / Collevecchio, Andrea; Hamza, Kais; Liu, Yunxuan.

Research output: Contribution to journal › Article › Research › peer-review

TY - JOUR

T1 - Invariance principle for biased bootstrap random walks

AU - Collevecchio, Andrea

AU - Hamza, Kais

AU - Liu, Yunxuan

PY - 2019/3

Y1 - 2019/3

N2 - Our main goal is to study a class of processes whose increments are generated via a cellular automata rule. Given the increments of a simple biased random walk, a new sequence of (dependent) Bernoulli random variables is produced. It is built, from the original sequence, according to a cellular automata rule. Equipped with these two sequences, we construct two more according to the same cellular automata rule. The construction is repeated a fixed number of times yielding an infinite array ({−K,…,K}×N) of (dependent) Bernoulli random variables. Taking partial sums of these sequences, we obtain a (2K+1)-dimensional process whose increments belong to the state space {−1,1}2K+1. The aim of the paper is to study the long term behaviour of this process. In particular, we establish transience/recurrence properties and prove an invariance principle. The limiting behaviour of these processes depends strongly on the direction of the iteration, and exhibits few surprising features. This work is motivated by an earlier investigation (see Collevecchio et al. (2015)), in which the starting sequence is symmetric, and by the related work Ferrari et al. (2000).

AB - Our main goal is to study a class of processes whose increments are generated via a cellular automata rule. Given the increments of a simple biased random walk, a new sequence of (dependent) Bernoulli random variables is produced. It is built, from the original sequence, according to a cellular automata rule. Equipped with these two sequences, we construct two more according to the same cellular automata rule. The construction is repeated a fixed number of times yielding an infinite array ({−K,…,K}×N) of (dependent) Bernoulli random variables. Taking partial sums of these sequences, we obtain a (2K+1)-dimensional process whose increments belong to the state space {−1,1}2K+1. The aim of the paper is to study the long term behaviour of this process. In particular, we establish transience/recurrence properties and prove an invariance principle. The limiting behaviour of these processes depends strongly on the direction of the iteration, and exhibits few surprising features. This work is motivated by an earlier investigation (see Collevecchio et al. (2015)), in which the starting sequence is symmetric, and by the related work Ferrari et al. (2000).

KW - Bootstrap random walks

KW - Functional limit theorem

KW - Random walks

UR - http://www.scopus.com/inward/record.url?scp=85046119887&partnerID=8YFLogxK

U2 - 10.1016/j.spa.2018.03.022

DO - 10.1016/j.spa.2018.03.022

M3 - Article

VL - 129

SP - 860

EP - 877

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

SN - 0304-4149

IS - 3

ER -