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January 2010 A CLT for the L2 modulus of continuity of Brownian local time
Xia Chen, Wenbo V. Li, Michael B. Marcus, Jay Rosen
Ann. Probab. 38(1): 396-438 (January 2010). DOI: 10.1214/09-AOP486

Abstract

Let $\{L_t^x; (x, t)∈R^1×R_+^1\}$ denote the local time of Brownian motion, and $$α_t:=∫_{−∞}^∞(L_t^x)^2 dx.$$ Let $η=N(0, 1)$ be independent of $α_t$. For each fixed $t$, $$\frac{\int_{-\infty}^{\infty}(L_{t}^{x+h}-L_{t}^{x})^{2}\,dx-4ht}{h^{3/2}}\stackrel{\mathcal{L}}{\rightarrow}\biggl(\frac{64}{3}\biggr)^{1/2}\sqrt{\alpha_{t}}\eta$$ as $h→0$. Equivalently, $$\frac{\int_{-\infty}^{\infty}(L^{x+1}_{t}-L^{x}_{t})^{2}\,dx-4t}{t^{3/4}}\stackrel{\mathcal{L}}{\rightarrow}\biggl(\frac{64}{3}\biggr)^{1/2}\sqrt{\alpha_{1}}\eta$$ as $t→∞$.

Citation

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Xia Chen. Wenbo V. Li. Michael B. Marcus. Jay Rosen. "A CLT for the L2 modulus of continuity of Brownian local time." Ann. Probab. 38 (1) 396 - 438, January 2010. https://doi.org/10.1214/09-AOP486

Information

Published: January 2010
First available in Project Euclid: 25 January 2010

MathSciNet: MR2599604
Digital Object Identifier: 10.1214/09-AOP486

Subjects:
Primary: 60F05, 60G17, 60J55

Rights: Copyright © 2010 Institute of Mathematical Statistics

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Vol.38 • No. 1 • January 2010
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