...as a function of NIG (Normal Inverse Gaussian) or VG (Variance Gamma) parameters, obviously. I've read that the NIG $\alpha$ is related to the $\alpha$-stable tail parameter, which conversely maps to the tail index.
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$\begingroup$ Please provide a definition of the tail index that you are interested in. In the NIG case, $\alpha$ is some kind of tail parameter. $\endgroup$– Richi WaJan 9, 2016 at 16:49
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$\begingroup$ The (Pareto) tail index is a very well known asymptotic tail decay parameter: $1 − F(x) \sim cx^{−\alpha}$ for $x\rightarrow\infty$, as estimated e.g. by the Hill or Pickands estimators. $\endgroup$– QuartzMar 8, 2016 at 10:42
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If the tail index is $\alpha$ then moments beyond that do not exist. The fact that all moments exist for NIG and VG indicates that the tail index is infinite (like the Gaussian).
