The Annals of Probability

A CLT for the L2 modulus of continuity of Brownian local time

Xia Chen, Wenbo V. Li, Michael B. Marcus, and Jay Rosen

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Let {Ltx; (x, t)∈R1×R+1} denote the local time of Brownian motion, and


Let η=N(0, 1) be independent of αt. For each fixed t,

\[\frac{\int_{-\infty}^{\infty}(L_{t}^{x+h}-L_{t}^{x})^{2}\,dx-4ht}{h^{3/2}}\stackrel{\mathcaligr{L}}{\rightarrow}\biggl(\frac{64}{3}\biggr)^{1/2}\sqrt{\alpha_{t}}\eta \]

as h→0. Equivalently,

\[\frac{\int_{-\infty}^{\infty}(L^{x+1}_{t}-L^{x}_{t})^{2}\,dx-4t}{t^{3/4}}\stackrel{\mathcaligr{L}}{\rightarrow}\biggl(\frac{64}{3}\biggr)^{1/2}\sqrt{\alpha_{1}}\eta \]

as t→∞.

Article information

Ann. Probab., Volume 38, Number 1 (2010), 396-438.

First available in Project Euclid: 25 January 2010

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Mathematical Reviews number (MathSciNet)

Primary: 60J55: Local time and additive functionals 60F05: Central limit and other weak theorems 60G17: Sample path properties

CLT Brownian local times modulus of continuity


Chen, Xia; Li, Wenbo V.; Marcus, Michael B.; Rosen, Jay. A CLT for the L 2 modulus of continuity of Brownian local time. Ann. Probab. 38 (2010), no. 1, 396--438. doi:10.1214/09-AOP486.

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