Long runs under a conditional limit distribution

TL;DR

This paper develops precise density approximations for long runs of a random walk conditioned on endpoint or average constraints, extending the Gibbs principle and enabling applications in rare event probability estimation and statistical inference.

Contribution

It introduces a sharp approximation method for the density of long conditioned random walk segments, extending the Gibbs conditional principle and providing tools for rare event analysis and inference.

Findings

Provides density approximations valid in large deviation regimes.

Extends Gibbs conditional principle to long subsequences.

Offers algorithms for simulation and approximation validity assessment.

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.