My question is how to compute R(dx). But before I can ask that I have to write down the background to my problem, so bear withme
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A tempered stable distribution is when a stable distribution is tempered by an exponential function of the form [latex]e^{-\theta{x}}[/latex]. In my particular case we are using a tempered stable law defined by Barndorff-Nielsen in the paper "modified stable processes" found here, http://economics.oul...nmsprocnew1.pdf.
In Barndorff's paper, [latex]\theta = (1/2)\gamma^{1/\alpha}[/latex], hence the tempering function is defined as [latex]e^{-(1/2} \gamma^{1/\alpha}{x}[/latex].
In Rosinski's paper on "tempering stable processes" (which can be found here or here) he states that tempering of the stable density [latex]f \mapsto f_{\theta}[/latex] leads to tempering of the corresponding Levy measure [latex]M \mapsto M_{\theta}[/latex], where [latex]M_{\theta}(dx) = e^{-\theta{x}}M(dx)[/latex].
Rosinski then goes on to say the Levy measure of a stable law in polar coordinates is of the form
[latex]M_0(dr, du) = r^{-\alpha-1}dr\sigma(du) \hspace{30mm} (2.1)[/latex]
and then says the Levy measure of a tempered stable density can be written as
[latex]M(dr, du) = r^{-\alpha-1}q(r,u)dr\sigma(du) \hspace{30mm} (2.2)[/latex]
he then says, the tempering function q in (2.2) can be represented as
[latex]q(r,u) = \int_0^{\infty}e^{-rs}Q(ds|u) \hspace{30mm} (2.3)[/latex]
Rosinski's paper also defines a measure R by
[latex]R(A) = \int_{R^d} I_A(x/||x||^2)||x||^{\alpha}Q(dx) \hspace{30mm} (2.5)[/latex]
and has
[latex]Q(A) = \int_{R^d} I_A(x/||x||^2)||x||^{\alpha}R(dx) \hspace{30mm} (2.6)[/latex]
now I know that for my particular tempered stable density the levy measure M is given by
[latex]2^{\alpha}\delta\frac{\alpha}{ \Gamma(1-\alpha)}x^{-1-\alpha}e^{-(1/2)\gamma^{1/\alpha}x}dx[/latex]
Rosinski then goes on to state Theorem 2.3: The Levy measure M of a tempered stable distribution can be written in the form
[latex]M(A)=\int_{R^d}\int_0^{\infty} I_A(tx)t^{-\alpha-1}e^{-t}dtR(dx)[/latex]
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So the question is, how can I work out what [latex]Q[/latex] is? and what is [latex]R(dx)[/latex]?
