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- W2038118712 abstract "Let $X={X_{t},tin R_{+}}$ 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 $1<betaleq 2$, and satisfies some additional regularity conditions, $$ lim_{ttoinfty}{ int_{-infty}^{infty} ( L^{ x+1}_{t}- L^{ x}_{ t})^{ 2},dx- Eleft(int_{-infty}^{infty} ( L^{ x+1}_{t}- L^{ x}_{ t})^{ 2},dxright)over tsqrt{psi^{-1}(1/t)}}$$ is equal in law to $$(8c_{psi,1 })^{1/2}left(int_{-infty}^{infty} left(L_{beta,1}^{x}right)^{2},dxright)^{1/2},eta$$ 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 $1<betaleq 2$ and satisfies some additional regularity conditions $$lim_{hto 0}sqrt{hpsi^{2}(1/h)} left{ int_{-infty}^{infty} ( L^{ x+h}_{1}- L^{ x}_{ 1})^{ 2},dx- Eleft( int_{-infty}^{infty} ( L^{ x+h}_{1}- L^{ x}_{ 1})^{ 2},dxright)right}$$ is equal in law to $$(8c_{beta,1})^{1/2},,eta,, left( int_{-infty}^{infty} (L_{1}^{x})^{2},dxright)^{1/2}$$ where $eta$ is a normal random variable with mean zero and variance one that is independent of ${L^{ x }_{ 1},xin R^{1}}$, and $c_{beta,1}$ is a known constant." @default.
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- W2038118712 date "2012-01-01" @default.
- W2038118712 modified "2023-10-18" @default.
- W2038118712 title "Central limit theorems for the $L^{2}$ norm of increments of local times of Lévy processes" @default.
- W2038118712 doi "https://doi.org/10.1214/ejp.v17-1740" @default.
- W2038118712 hasPublicationYear "2012" @default.
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