A stochastic calculus proof of the CLT for the L^{2} modulus of continuity of local time

TL;DR

This paper provides a stochastic calculus-based proof of the Central Limit Theorem for the L^{2} modulus of continuity of Brownian local time, demonstrating convergence to a normal distribution as the parameter h approaches zero.

Contribution

It introduces a novel stochastic calculus approach to establish the CLT for the L^{2} modulus of continuity of local time, expanding the theoretical understanding of local time fluctuations.

Findings

Proves the CLT for the L^{2} modulus of continuity of local time.

Shows convergence in distribution to a normal variable as h tends to zero.

Provides a stochastic calculus proof method for local time analysis.

Abstract

We give a stochastic calculus proof of the Central Limit Theorem \[ {\int (L^{x+h}_{t}- L^{x}_{t})^{2} dx- 4ht\over h^{3/2}} \stackrel{\mathcal{L}}{\Longrightarrow}c(\int (L^{x}_{t})^{2} dx)^{1/2} \eta\] as $h\to 0$ for Brownian local time $L^{x}_{t}$. Here $\eta$ is an independent normal random variable with mean zero and variance one.