Kac-Stroock type approximations for the Brownian motion

TL;DR

The paper extends Kac-Stroock-type approximations of Brownian motion by replacing the Poisson driver with a general renewal process $L$ whose inter-arrival times $U_k$ possess a finite $p$-th moment for some $p>2$. It shows that, with $X_n(t)= C\sqrt{n}\int_0^t (-1)^{L(nu)}du$ and $C^2=\mathbb{E}[U_1]/\mathrm{Var}(U_1)$, the processes converge weakly to standard Brownian motion in $\mathcal{C}([0,1])$, generalizing Stroock's 1982 result and aligning the phenomenon with Donsker's invariance principle. The proof hinges on a decomposition $X_n = W_n+R_n$, proving $R_n\to 0$ in the Skorokhod sense and $W_n\to B$ by reducing to i.i.d. differences $U_{2j-1}-U_{2j}$ and applying the classical invariance principle with a suitable time-change. This work broadens the class of renewal processes that yield Brownian limits and clarifies the connection to random-walk limits through time-change techniques.

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.

Theorems & Definitions (2)

• Theorem 2.1
• proof : Proof of Theorem \ref{['main theorem']}