Long runs under point conditioning. The real case

TL;DR

This paper develops a precise approximation for the distribution of long runs of a conditioned random walk, extending the Gibbs principle, and provides algorithms for simulation and length determination of such runs.

Contribution

It introduces a sharp density approximation for long conditioned random walks and algorithms for simulation and maximal length determination.

Findings

Provides a sharp approximation for the density of long conditioned runs.

Extends the Gibbs conditional principle to long subsequences.

Includes algorithms for simulation and length determination.

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 functions of its summands as their number tends to infinity. The conditioning event is of moderate or large deviation type. The result extends the Gibbs conditional principle in the sense that it provides a description of the distribution of the random walk on long subsequences. An algorithm for the simulation of such long runs is presented, together with an algorithm determining their maximal length for which the approximation is valid up to a prescribed accuracy.