## The Monte Carlo Computation Error of Transition Probabilities

Please always quote using this URN: urn:nbn:de:0297-zib-59933

- In many applications one is interested to compute transition probabilities of a Markov chain. This can be achieved by using Monte Carlo methods with local or global sampling points. In this article, we analyze the error by the difference in the $L^2$ norm between the true transition probabilities and the approximation achieved through a Monte Carlo method. We give a formula for the error for Markov chains with locally computed sampling points. Further, in the case of reversible Markov chains, we will deduce a formula for the error when sampling points are computed globally. We will see that in both cases the error itself can be approximated with Monte Carlo methods. As a consequence of the result, we will derive surprising properties of reversible Markov chains.

Author: | Adam Nielsen |
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Document Type: | ZIB-Report |

Tag: | Computation error; Markov operator; Measurable state space; Monte Carlo methods; Reversible Markov chain |

MSC-Classification: | 37-XX DYNAMICAL SYSTEMS AND ERGODIC THEORY [See also 26A18, 28Dxx, 34Cxx, 34Dxx, 35Bxx, 46Lxx, 58Jxx, 70-XX] |

65-XX NUMERICAL ANALYSIS | |

CCS-Classification: | G. Mathematics of Computing / G.1 NUMERICAL ANALYSIS |

G. Mathematics of Computing / G.3 PROBABILITY AND STATISTICS | |

PACS-Classification: | 00.00.00 GENERAL / 02.00.00 Mathematical methods in physics |

Date of first Publication: | 2016/07/22 |

Series (Serial Number): | ZIB-Report (16-37) |

ISSN: | 1438-0064 |

Published in: | appeared in: Statistics & Probability Letters 118 (2016) pp. 163-170. |