Gaussian approximation of multivariate Lévy processes with applications to simulation of tempered stable processes (Q880482)

The paper is devoted to the problem of the Gaussian of a Lévy process. If \((X_\varepsilon(t))_{t\geq 0}\) is a Lévy process with corresponding Lévy measure \(\nu_\varepsilon (dx)\), \[ E^{i\langle y,X_\varepsilon(t)\rangle} =\exp \left\{ t\int_{\mathbb{R}^n} [e^{i\langle y,x\rangle} -1-\langle y,x\rangle]\,\nu_\varepsilon (dx)\right\}, \] where \( \int_{\mathbb{R}^n} \| x\| ^2\,\nu_\varepsilon (dx)<\infty. \) In this case \(X_\varepsilon(t)\) has zero mean and covariance matrix \(t\Sigma_\varepsilon=t \int_{\mathbb{R}^n} x\,x^T\, \nu_\varepsilon (dx)\). If \(\Sigma_\varepsilon\) is non-singular for every \(\varepsilon \in(0,1]\), necessary and sufficient conditions for the weak convergence in the Skorokhod space \(\Sigma_\varepsilon^{-1/2} X_\varepsilon \to W\), where \(W\) is a standard Brownian motion in \(\mathbb{R}^n\), are given, and conditions, when the sufficient part is satisfied, are provided. This result gives a way how to approximate a multivariant Lévy process. Namely, let \(X(t)\) be a Lévy process determined by the characteristic function \[ E^{i\langle y,X(t)\rangle} =\exp \left\{ t\left[ i\langle a,y\rangle +\int_{\mathbb{R}^n} [e^{i\langle y,x\rangle} -1-\langle y,x\rangle (\| x\| \leq 1)]\,\nu (dx)\right] \right\}. \] Suppose we can decompose \(\nu\) by \( \nu=\nu_\varepsilon+\nu^\varepsilon, \) where \(\nu_\varepsilon\) satisfies \( \int_{\mathbb{R}^n} \| x\| ^2\,\nu_\varepsilon (dx)<\infty\) and \(\nu^\varepsilon(\mathbb{R}^n)<\infty.\) The authors find conditions, under which \(X\) can be decomposed into the sum of compound Poisson processes with jump measure \(\nu^\varepsilon\), a drift \(a_\varepsilon\), \(A_\varepsilon W\), where \(A_\varepsilon\) is some non-singular matrix, and some càdlàg process \(Y_\varepsilon\). This method is illustrated on multivariant stable and tempered stable processes.