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December 2014 Long runs under a conditional limit distribution
Michel Broniatowski, Virgile Caron
Ann. Appl. Probab. 24(6): 2246-2296 (December 2014). DOI: 10.1214/13-AAP975

Abstract

This paper presents a sharp approximation of the density of long runs of a random walk conditioned on its end value or by an average of a function of its summands as their number tends to infinity. In the large deviation range of the conditioning event it extends the Gibbs conditional principle in the sense that it provides a description of the distribution of the random walk on long subsequences. An approximation of the density of the runs is also obtained when the conditioning event states that the end value of the random walk belongs to a thin or a thick set with a nonempty interior. The approximations hold either in probability under the conditional distribution of the random walk, or in total variation norm between measures. An application of the approximation scheme to the evaluation of rare event probabilities through importance sampling is provided. When the conditioning event is in the range of the central limit theorem, it provides a tool for statistical inference in the sense that it produces an effective way to implement the Rao–Blackwell theorem for the improvement of estimators; it also leads to conditional inference procedures in models with nuisance parameters. An algorithm for the simulation of such long runs is presented, together with an algorithm determining the maximal length for which the approximation is valid up to a prescribed accuracy.

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Michel Broniatowski. Virgile Caron. "Long runs under a conditional limit distribution." Ann. Appl. Probab. 24 (6) 2246 - 2296, December 2014. https://doi.org/10.1214/13-AAP975

Information

Published: December 2014
First available in Project Euclid: 26 August 2014

zbMATH: 1350.60006
MathSciNet: MR3262503
Digital Object Identifier: 10.1214/13-AAP975

Subjects:
Primary: 60B10
Secondary: 65C50

Keywords: Conditioned random walk , Gibbs principle , importance sampling , large deviation , Moderate deviation , Rao–Blackwell theorem , simulation

Rights: Copyright © 2014 Institute of Mathematical Statistics

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Vol.24 • No. 6 • December 2014
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