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

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