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It is well known that that the gamma and inverse gaussian distributions lead to Levy processes.

For a Levy process with Levy measure $\nu$ one can define (I'm new to this, this is the first definition I have seen) the Blumenthal-Getoor index $$ \beta = \inf \{p >0 : \int_{|x|\leq 1 } |x|^p \nu(dx) < \infty \} $$ Looking in the class of jump processes I know that $\beta=0$ gives compound poisson $\beta = 1$ finite activity and necessarily $\beta \leq 2$ - so I guess it can generally be thought of as a kind of measure of how "wild" the process behaves.

Edit the paper I have tried to guess the definition from is this on page 4, as I have never heard of it before I did not know it was incorrect, I have changed it to another definition (from google).

What I am looking for is whether this number is known for the Inverse Gaussian Levy process, the Gamma Levy process and maybe a reference to get a feeling for this new object.

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  • $\begingroup$ For the Gamma process, the Lévy measure is $\nu(dx) = x^{-1} e^{-x} dx$ for $x\geq 0$. From your definition of the Blumenthal-Getoor index, you see that $\beta = 0$. The difference with the compound Poisson case is that the infimum $0$ is not attain. $\endgroup$ – Goulifet Feb 17 '15 at 13:35
  • $\begingroup$ Small question: in the definition of $\beta$, shouldn't the integral be over all $\mathbb{R}$? $\endgroup$ – Goulifet Feb 17 '15 at 14:25
  • $\begingroup$ As it is written now, the definition of the Blumenthal-Getoor index doesn't make sense; there are several typos in it. $\endgroup$ – saz Feb 17 '15 at 14:28
  • $\begingroup$ @saz Thanks for pointing out - please see edit. $\endgroup$ – Henrik Feb 17 '15 at 16:00
  • $\begingroup$ @Henrik yeah, looks much better. See my answer below (... if you are interested in more details, I can try to find some references; the references I know are unfortunately for more general processes and that might be confusing.) $\endgroup$ – saz Feb 17 '15 at 16:29
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Example 1 (Gamma process): As @Goulifet pointed out, the Lévy measure of a Gamma process is given by $$\nu(dx) =c x^{-1} e^{-\alpha x} \, dx =: p(x) \, dx.$$ Here $\alpha$ and $c$ are fixed constants. Since $e^{-\alpha x} \leq e^{\alpha}$ for all $x \in [-1,1]$, we get

$$\int_{|x| \leq 1} |x|^p \, \nu(dx) \leq c e^{\alpha} \int_{|x| \leq 1} |x|^{p-1} \, dx < \infty$$ for all $p>0$. Consequently, $\beta=0$.

Example 2 (Inverse Gaussian process): The density of the Lévy measure of the inverse Gaussian distribution is given by

$$p(x) = c \frac{1}{x^{3/2}} e^{-\gamma^2 x/2}$$

for some constant $c>0$ and $\gamma>0$. Using the same argumentation as above, we find that

$$\int_{|x| \leq 1} |x|^p \, \nu(dx) \leq c e^{\gamma^2/2} \int_{|x| \leq 1} |x|^{p-3/2}< \infty$$

for all $p>1/2$. Consequently, $\beta=1/2$.

General remarks: It is well-known that a Lévy process $(X_t)_{t \geq 0}$ can be characterized by the Lévy triplet $(b,\sigma^2,\nu)$ as well its symbol $\psi$. Namely, $\psi$ satisfies $$\mathbb{E}e^{\imath \, \xi X_t} = e^{- t \psi(\xi)}.$$ By definition, the Blumenthal-Getoor index measures the intensity of small jumps. Roughly speaking: If $(X_t)_{t \geq 0}$ and $(Y_t)_{t \geq 0}$ are Lévy processes with Blumenthal-Getoor indizes $\beta$ and $\gamma$, respectively, and if $\beta>\gamma$, then "$(X_t)_{t \geq 0}$ has more small jumps than $(Y_t)_{t \geq 0}$". One can show that the Blumenthal-Getoor index can be also characterized in terms of the asymptotic behavior of $\psi$ as $\xi \to \infty$: $$\beta = \inf \left\{\lambda>0; \lim_{|\xi| \to \infty} \frac{|\psi(\xi)|}{|\xi|^{\lambda}}=0\right\}$$ if the Lévy triplet satisfies a so-called weak sector condition (i.e. $|\text{Im}\ \psi(\xi)| \leq C |\text{Re}\ \psi(\xi)|$ for some constant $C>0$) and $\sigma^2=0$.

There is a lot of interest in Blumenthal-Getoor indizes because they can be used to characterize path properties (e.g. Hausdorff dimension and asymptotics of $\sup_{s \leq t} |X_s|$) as well as the asymptotics of absolute moments.

Some References: The following two references are concerned with Blumenthal-Getoor-Indizes for so-called Feller processes (this class includes Lévy processes).

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  • $\begingroup$ Henrik defined the Blumenthal-Getoor index in terms of the Lévy measure. In your general remark, you express it with the assymptotic behavior of $\psi$, then you suggest that the two concepts are linked with a "weak sector condition". Can you tell me more about it? In particular, when are the definitions with the Lévy measure and the symbol (with $\sigma^2=0$) equivalent? $\endgroup$ – Goulifet Mar 9 '15 at 19:10
  • $\begingroup$ @Goulifet Sector condition means that $|\text{Im} \psi(\xi)| \leq C |\text{Re} \, \psi(\xi)|$ for some constant $C>0$. If this condition holds, then both characterizations are equivalent. $\endgroup$ – saz Mar 9 '15 at 19:21
  • $\begingroup$ thanks a lot for your answer. Do you have any good reference addressing this kind of question in general? $\endgroup$ – Goulifet Mar 10 '15 at 8:44
  • $\begingroup$ @Goulifet What do you mean by "in general"? For a larger class of processes? $\endgroup$ – saz Mar 10 '15 at 9:27
  • $\begingroup$ I mean: references showing for instance that the sector condition is sufficient (and necessary?) to have indices from the Lévy measure and the symbol equal. And more generally, discussing about Blumenthal-Getoor indices and their applications. I know some quite old papers (from the 70's), but I have never find a general exposition on the subject. $\endgroup$ – Goulifet Mar 10 '15 at 9:34
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In this paper, you can find the Lévy measures associated with both the Inverse Gaussian Lévy process (Example 3.2) and the Gamma Lévy process (Example 3.1).

For the Gamma process, I already mentioned that $\beta =0$. For the Inverse Gaussian, the Lévy measure has a density $u(x) \propto x^{-3/2} e^{- x}$, also you obtain $\beta = 1/2$.

The index $\beta$ can be described as a measure of the local regularity of a Lévy process. In the original paper of Blumenthal and Getoor, I think it was introduced to study the behavior of a Lévy process at $0$ (how does it converge to $0$?). I also mention that paper as an illustration of the link between the local regularity of a Lévy process and $\beta$. In the Theorem of page 5, the local Besov regularity (in the paper, the regularity parameter is $s$) is measured in terms of $\beta_\infty$ (what you call $\beta$).

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