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

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

2009-07-15

arxiv.org/abs/0907.2693v2

Mathematics

Probability

Scientific paper

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

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