A CLT for the L^{2} modulus of continuity of Brownian local time

TL;DR

This paper establishes a central limit theorem for the L^{2} modulus of continuity of Brownian local time, revealing its asymptotic distribution as the parameter approaches zero or infinity.

Contribution

It introduces a new CLT for the L^{2} modulus of continuity of Brownian local time, connecting it to the local time’s quadratic variation and Gaussian limits.

Findings

Convergence in distribution to a Gaussian limit involving local time.

Asymptotic behavior of the L^{2} modulus of continuity as h→0.

Asymptotic behavior of the L^{2} modulus of continuity as t→∞.

Abstract

Let $\{L^{x}_{t} ; (x,t)\in R^{1}\times R^{1}_{+}\}$ denote the local time of Brownian motion and \[ \alpha_{t}:=\int_{-\infty}^{\infty} (L^{x}_{t})^{2} dx . \] Let $\eta=N(0,1)$ be independent of $\alpha_{t}$. For each fixed $t$ \[ {\int_{-\infty}^{\infty} (L^{x+h}_{t}- L^{x}_{t})^{2} dx- 4ht\over h^{3/2}} \stackrel{\mathcal{L}}{\to}({64 \over 3})^{1/2}\sqrt{\alpha_{t}} \eta, \] as $h\rar 0$. Equivalently \[ {\int_{-\infty}^{\infty} (L^{x+1}_{t}- L^{x}_{t})^{2} dx- 4t\over t^{3/4}} \stackrel{\mathcal{L}}{\to}({64 \over 3} )^{1/2}\sqrt{\alpha_{1}} \eta, \] as $t\rar\infty$.