Long runs under a conditional limit distribution

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.Comment: 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:1010.361

Small Variance Estimators for Rare Event Probabilities

“…However the above theorem provides optimal approximations on the entire space R k for all k between 1 and n − 1 in the gaussian case and u(x) = x, since g ns y k 1 coincides with the conditional density. As stated in [Broniatowski and Caron 2011], the extension of our results from typical paths to the whole space R k holds: convergence of the relative error on large sets imply that the total variation distance between the conditioned measure and its approximation goes to 0 on the entire space. So our results provide an extension of [Diaconis and Freedman 1988] and [Dembo and Zeitouni (1996)] who considered the case when k is of small order with respect to n; the conditions which are assumed in the present paper are weaker than those assumed in the just cited works; however, in contrast with their results, we do not provide explicit rates for the convergence to 0 of the total variation distance on R k .…”

“…when v belongs to (a, ∞) . We refer to [Broniatowski and Caron 2011] for this result. We introduce a positive sequence n which satisfies…”

“…In [Broniatowski and Caron 2011], a similar question is addressed and a proxy of the curve δ → k δ is provided in order to define the maximal k leading to a given relative accuracy under the point condition (U 1,n = na) , namely when p nA is replaced by p na and g nA by g na .…”

Some Recent Results in Rare Event Estimation

“…The IS density on R n is obtained multiplying this proxy by a product of a much simpler adaptive IS scheme following [5]. We rely on [7] where the basic approximation used in the present section can be found. All proofs of the results in the present section can be found in [8].…”

Importance Sampling for Multi-Constraints Rare Event Probability

“…We rely on [7] where the basic approximation (and proofs) used in the present paper can be found. The real case is studied in [4] and applications for IS estimators can be found in [3].…”