Blumenthal-Getoor index for IG and $\Gamma$

Question

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.

Answer 1

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).

• René L. Schilling: Growth and Hölder conditions for the sample paths of Feller processes, Probability Theory and Related Fiels, 1998; Chapter 5 (Proof of the above mentioned equivalent characterization)
• Björn Böttcher, René L. Schilling, Jian Wang: Lévy Matters III, Springer 2014; Section 5.2+5.3 (Applications)

Answer 2

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 $\beta$).