## Abstract

In this article we study a small random perturbation of a linear recurrence

equation. If all the roots of its corresponding characteristic equation

have modulus strictly less than one, the random linear recurrence goes exponentially

fast to its limiting distribution in the total variation distance as time

increases. By assuming that all the roots of its corresponding characteristic

equation have modulus strictly less than one and rather mild conditions, we

prove that this convergence happens as a switch-type, i.e., there is a sharp

transition in the convergence to its limiting distribution. This fact is known as

a cut-off phenomenon in the context of stochastic processes.

equation. If all the roots of its corresponding characteristic equation

have modulus strictly less than one, the random linear recurrence goes exponentially

fast to its limiting distribution in the total variation distance as time

increases. By assuming that all the roots of its corresponding characteristic

equation have modulus strictly less than one and rather mild conditions, we

prove that this convergence happens as a switch-type, i.e., there is a sharp

transition in the convergence to its limiting distribution. This fact is known as

a cut-off phenomenon in the context of stochastic processes.

Original language | English |
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Journal | Brazilian Journal of Probability and Statistics |

Volume | 35 |

Issue number | 2 |

Pages (from-to) | 224-241 |

Number of pages | 18 |

ISSN | 0103-0752 |

DOIs | |

Publication status | Published - May 2021 |

Externally published | Yes |

MoE publication type | A1 Journal article-refereed |

## Fields of Science

- 111 Mathematics
- Linear recurrences
- 112 Statistics and probability
- Cut-off phenomenon
- Gaussian distribution
- Total variation distance
- linear recurrences
- Gaussian distribution
- total variation dis-tance
- &
- nbsp
- Cut-off phenomenon