A CLT for the third integrated moment of Brownian local time increments

TL;DR

This paper establishes a central limit theorem for the third integrated moment of Brownian local time increments, extending previous results for the second moment and highlighting limitations for higher moments.

Contribution

The paper introduces a CLT for the third moment of Brownian local time increments, generalizing earlier second-moment results and discussing methodological limitations.

Findings

Asymptotic normality of the third integrated moment of local time increments.

Explicit variance formula involving the third moment of local time.

Discussion on why the approach does not extend to higher moments.

Abstract

Let $\{L^{x}_{t} ; (x,t)\in R^{1}\times R^{1}_{+}\}$ denote the local time of Brownian motion. Our main result is to show that for each fixed $t$ $${\int (L^{x+h}_t- L^x_t)^3 dx-12h\int (L^{x+h}_t - L^x_t)L^x_t dx-24h^{2}t\over h^2} \stackrel{\mathcal{L}}{\Longrightarrow}\sqrt{192}(\int (L^x_t)^3dx)^{1/2}\eta$$ as $h\to 0$, where $\eta$ is a normal random variable with mean zero and variance one that is independent of $L^{x}_{t}$. This generalizes our previous result for the second moment. We also explain why our approach will not work for higher moments