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2012 Central limit theorems for the $L^{2}$ norm of increments of local times of Lévy processes
Michael Marcus, Jay Rosen
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Electron. J. Probab. 17: 1-111 (2012). DOI: 10.1214/EJP.v17-1740

Abstract

Let $X=\{X_{t},t\in R_{+}\}$ be a symmetric Lévy process with local time $\{L^{ x }_{ t}\,;\,(x,t)\in R^{ 1}\times R^{ 1}_{ +}\}$. When the Lévy exponent $\psi(\lambda)$ is regularly varying at zero with index $1<\beta\leq 2$, and satisfies some additional regularity conditions, $$ \lim_{t\to\infty}{ \int_{-\infty}^{\infty} ( L^{ x+1}_{t}- L^{ x}_{ t})^{ 2}\,dx- E\left(\int_{-\infty}^{\infty} ( L^{ x+1}_{t}- L^{ x}_{ t})^{ 2}\,dx\right)\over t\sqrt{\psi^{-1}(1/t)}}$$ is equal in law to $$(8c_{\psi,1 })^{1/2}\left(\int_{-\infty}^{\infty} \left(L_{\beta,1}^{x}\right)^{2}\,dx\right)^{1/2}\,\eta$$ where $L_{\beta,1}=\{L^{ x }_{\beta, 1}\,;\, x \in R^{ 1} \}$ denotes the local time, at time 1, of a symmetric stable process with index $\beta$, $\eta$ is a normal random variable with mean zero and variance one that is independent of $L _{ \beta,1}$, and $c_{\psi,1}$ is a known constant that depends on $\psi$.When the Lévy exponent $\psi(\lambda)$ is regularly varying at infinity with index $1<\beta\leq 2$ and satisfies some additional regularity conditions $$\lim_{h\to 0}\sqrt{h\psi^{2}(1/h)} \left\{ \int_{-\infty}^{\infty} ( L^{ x+h}_{1}- L^{ x}_{ 1})^{ 2}\,dx- E\left( \int_{-\infty}^{\infty} ( L^{ x+h}_{1}- L^{ x}_{ 1})^{ 2}\,dx\right)\right\}$$ is equal in law to $$(8c_{\beta,1})^{1/2}\,\,\eta\,\, \left( \int_{-\infty}^{\infty} (L_{1}^{x})^{2}\,dx\right)^{1/2}$$ where $\eta$ is a normal random variable with mean zero and variance one that is independent of $\{L^{ x }_{ 1},x\in R^{1}\}$, and $c_{\beta,1}$ is a known constant.

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Michael Marcus. Jay Rosen. "Central limit theorems for the $L^{2}$ norm of increments of local times of Lévy processes." Electron. J. Probab. 17 1 - 111, 2012. https://doi.org/10.1214/EJP.v17-1740

Information

Accepted: 18 January 2012; Published: 2012
First available in Project Euclid: 4 June 2016

zbMATH: 1246.60038
MathSciNet: MR2878786
Digital Object Identifier: 10.1214/EJP.v17-1740

Subjects:
Primary: 60F05
Secondary: 60G51 , 60J55

Keywords: $L^{2}$ norm of increments , central limit theorem , L\'evy process , Local time

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Vol.17 • 2012
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