Exchangeable stable random vectors and their simulations (Q5943410)

The simulation of Exchangeable Stable Random Vectors (ESRV) is considered. A random vector \(X= (X_1, \ldots, X_d)'\) is exchangeable if for every one of the \(d!\) permutation \(\{ \pi_1, \ldots, \pi_d\}\) of \(\{ 1, \ldots, d\}\) the random vectors \((X_1, \ldots, X_d)'\) and \((X_{\pi_1}, \ldots, X_{\pi_d})'\) have the same distribution. The random vector \(X= (X_1, \ldots, X_d)'\) is \(\alpha\)-stable if its characteristic function is given by \[ \phi(t)= \begin{cases} \exp\biggl\{ -\int_{S_{d-1}} \psi(\langle t, s\rangle) \Gamma(ds) +i \langle t, \mu\rangle\biggr\}, \;0<\alpha <2,\\ \exp\bigl\{ -\langle At, t\rangle +i \langle t, \mu\rangle \bigr\}, \;\alpha =2,\end{cases} \] where \[ \psi(t)= \begin{cases} |u|^\alpha \bigl(1 -i {\text sign}(u) \tan\bigl({\pi \alpha\over 2}\bigr)\bigr), 0<\alpha <2, \alpha \neq 1, \\ |u |\bigl( 1 + i{2\over \pi} {\text sign}(u) \ln(|u|)\bigr), \alpha =1,\end{cases} \] \(S_{d-1} = \{ s\in R^d\colon \|s\|=1\}\), \(\Gamma\) is a finite Borel measure on \(S_{d-1}.\) \(\Gamma\) is called the spectral measure of \(X.\) The authors present some characterization of an ESRV. For example, the \(\alpha\)-stable random vector \(X\), \(0< \alpha <2\), is exchangeable if and only if \(\mu_X\) has the same components and \(\Gamma_X\) is an exchangeable spectral measure that is \(\Gamma_X (E) = \Gamma_X(\pi E)\). The following result is proved. Let \(X\) be a ESRV, \(0< \alpha <2\). Then there exists a sequence of ESRV \(\{ X_m\}_{m=1}^\infty\) each having discrete spectral measure with finite number of atom \(S\) for which \(X_m {\buildrel d \over \longrightarrow} X_0.\) These assertions give a possibility to construct a simulation procedure for ESRV.