Kac-Stroock type approximations for the Brownian motion

TL;DR

This paper demonstrates that certain processes based on renewal processes with specific inter-arrival distributions converge weakly to Brownian motion, extending Stroock's earlier results with Poisson processes using Donsker's invariance principle.

Contribution

It generalizes Stroock's approximation of Brownian motion by replacing the Poisson process with more general renewal processes under integrability conditions.

Findings

Processes converge weakly to Brownian motion in continuous function space.

Generalizes Stroock's result to broader renewal processes.

Relies on Donsker's invariance principle for proof.

Abstract

In the present paper we show that the processes $X_n = \{X_n(t) \colon t \in [0,1]\}$, $n \in \mathbb{N}$, defined by $X_n(t) = \sqrt{n}C\int_0^t (-1)^{L(nu)} du$, where $L = \{L(t) \colon t \geq 0\}$ is a renewal processes whose inter-arrival times satisfy some integrability conditions and $C > 0$ is some normalizing constant, weakly converge, in the space of continuous functions over $[0,1]$, $\mathcal{C}([0,1])$, to the Brownian motion as $n$ approaches infinity. Thus, generalizing the result of D. W. Stroock (1982), where $L$ is taken to be a standard Poisson process. In particular, we see that these results are a mere consequence of Donsker's invariance principle.