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

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

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

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

Abstract

Let {L^_tx; (x, t)∈R¹×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,

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

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→∞.

First Page: Show

Primary Subjects: 60J55, 60F05, 60G17

Keywords: CLT; Brownian local times; modulus of continuity

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aop/1264434003
Digital Object Identifier: doi:10.1214/09-AOP486
Mathematical Reviews number (MathSciNet): MR2599604

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