### Abstract

Let {Wt}∞t=1 be a finite state stationary Markov chain, and suppose that f is a real-valued function on the state space. If f is bounded, then Gillman's expander Chernoff bound (1993) provides concentration estimates for the random variable f (W1) + · · · +f (Wn) that depend on the spectral gap of the Markov chain and the assumed bound on f. Here we obtain analogous inequalities assuming only that the q'th moment of f is bounded for some q ≥ 2. Our proof relies on reasoning that differs substantially from the proofs of Gillman's theorem that are available in the literature, and it generalizes to yield dimension-independent bounds for mappings f that take values in an Lp(μ) for some p ≥ 2, thus answering (even in the Hilbertian special case p = 2) a question of Kargin (Ann. Appl. Probab. 17 (4) (2007) 1202-1221).

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

Pages (from-to) | 2270-2280 |

Number of pages | 11 |

Journal | Annales de l'institut Henri Poincare (B) Probability and Statistics |

Volume | 56 |

Issue number | 3 |

DOIs | |

State | Published - Aug 2020 |

### Keywords

- Concentration bounds
- Expander graphs
- Gillman's theorem
- Markov chains

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

## Fingerprint Dive into the research topics of 'Concentration of Markov chains with bounded moments'. Together they form a unique fingerprint.

## Cite this

*Annales de l'institut Henri Poincare (B) Probability and Statistics*,

*56*(3), 2270-2280. https://doi.org/10.1214/19-AIHP1039