2012/02/29 by Michel Broniatowski, Virgile Caron · 3 citations
Decision Sciences · Mathematics · #Central limit theorem #Conditional probability #Conditional probability distribution #Conditioning #Event (particle physics) #Nuisance parameter #Probabilistic and Robust Engineering Design #Probability and Risk Models #Probability density function #Random walk #Range (aeronautics) #Risk and Portfolio Optimization #Sampling distribution #math.PR
paper · pdf · doi:10.1214/13-aap975
published in The Annals of Applied Probability 24(6) (Institute of Mathematical Statistics) · Published in at http://dx.doi.org/10.1214/13-AAP975 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org). arXiv admin note: text overlap with arXiv:1010.3616
openalex publication_date 2014/08/26 · arxiv created 2014/09/05 · arxiv updated 2014/09/08 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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.