### Abstract

We prove explicit error bounds for Markov chain Monte Carlo (MCMC) methods to compute expectations of functions with unbounded stationary variance. We assume that there is a p is an element of (1, 2) so that the functions have finite L-p-norm. For uniformly ergodic Markov chains we obtain error bounds with the optimal order of convergence n(1/p-1) and if there exists a spectral gap we almost get the optimal order. Further, a burn-in period is taken into account and a recipe for choosing the burn-in is provided. (C) 2015 Elsevier B.V. All rights reserved.

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

Pages (from-to) | 6-12 |

Journal | Statistics & probability letters |

Volume | 99 |

DOIs | |

Publication status | Published - Apr 2015 |

Externally published | Yes |

### Keywords

- Markov chain Monte Carlo
- Absolute mean error
- Uniform ergodicity
- Spectral gap

### Cite this

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*Statistics & probability letters*, vol. 99, pp. 6-12. https://doi.org/10.1016/j.spl.2014.07.035

**Error bounds of MCMC for functions with unbounded stationary variance.** / Rudolf, Daniel; Schweizer, Nikolaus.

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

TY - JOUR

T1 - Error bounds of MCMC for functions with unbounded stationary variance

AU - Rudolf, Daniel

AU - Schweizer, Nikolaus

PY - 2015/4

Y1 - 2015/4

N2 - We prove explicit error bounds for Markov chain Monte Carlo (MCMC) methods to compute expectations of functions with unbounded stationary variance. We assume that there is a p is an element of (1, 2) so that the functions have finite L-p-norm. For uniformly ergodic Markov chains we obtain error bounds with the optimal order of convergence n(1/p-1) and if there exists a spectral gap we almost get the optimal order. Further, a burn-in period is taken into account and a recipe for choosing the burn-in is provided. (C) 2015 Elsevier B.V. All rights reserved.

AB - We prove explicit error bounds for Markov chain Monte Carlo (MCMC) methods to compute expectations of functions with unbounded stationary variance. We assume that there is a p is an element of (1, 2) so that the functions have finite L-p-norm. For uniformly ergodic Markov chains we obtain error bounds with the optimal order of convergence n(1/p-1) and if there exists a spectral gap we almost get the optimal order. Further, a burn-in period is taken into account and a recipe for choosing the burn-in is provided. (C) 2015 Elsevier B.V. All rights reserved.

KW - Markov chain Monte Carlo

KW - Absolute mean error

KW - Uniform ergodicity

KW - Spectral gap

U2 - 10.1016/j.spl.2014.07.035

DO - 10.1016/j.spl.2014.07.035

M3 - Article

VL - 99

SP - 6

EP - 12

JO - Statistics & probability letters

JF - Statistics & probability letters

SN - 0167-7152

ER -