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.

  • $\begingroup$ 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

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).

  • $\begingroup$ 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

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 $\beta$).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.