Invariance principles for integrated random walks conditioned to stay positive

TL;DR

This paper establishes invariance principles and functional convergence results for integrated random walks conditioned to stay positive, extending classical results to this specific constrained process.

Contribution

It introduces invariance principles and convergence results for the meander and bridge of integrated random walks conditioned to remain positive, including their Doob's $h$-transform.

Findings

Invariance principles for the meander and bridge of integrated random walks.

Functional convergence of the Doob's $h$-transform to the Kolmogorov diffusion.

Extension of classical invariance principles to integrated random walks with positivity constraints.

Abstract

Let $S(n)$ be a centered random walk with finite second moment. We consider the integrated random walk $T(n) = S(0)+S(1)+\dots+S(n)$. We prove invariance principles for the meander and for the bridge of this process, under the condition that the integrated random walk remains positive. Furthermore, we prove the functional convergence of its Doob's $h$-transform to the $h$-transform of the Kolmogorov diffusion conditioned to stay positive.