Background:
I should probably start by saying I am an autodidact in almost everything pure maths.
I have two questions regarding the Levy-Khintchine representation; from Sato [1]:
If $\mu$ is an infinitely divisible distribution (IDD) on $\mathbb{R}^{d}$, then its characteristic function can be decomposed as
$$
\log(\hat\mu) = -\frac{1}{2}\langle \bullet, A\bullet\rangle+i\langle\gamma,\bullet\rangle + \int\limits_{\mathbb{R}^{d}}e^{i\langle \bullet, x\rangle}-1-i\langle \bullet,x\rangle \large{\unicode{x1D7D9}}_{||x||\leq1}(x) ~d\nu(x)
$$
where $A$ is a symmetric nonnegative-definite $d\times d$ matrix, $\gamma\in\mathbb{R}^{d}$, and $\nu$ is a measure satisfying
$$
\nu(\{0\})=0 ~~~\textit{and}~~~ \int\limits_{\mathbb{R}^{d}}\min(||x||_{2}^{2},1) ~d\nu(x) < \infty
$$
I'm looking for an abstract, measure-theoretic explanation of the following problems. In the lectures I am building, I have not introduced random variables, but am rather motivating the measure theory.
Question 1: What is the meaning of $i\langle z,x\rangle \large{\unicode{x1D7D9}}_{||\bullet||\leq1}(x)$ in the representation? I can get on board with the representation being the convolution of a Gaussian and Poisson, but the aforementioned term throws me off.
Question 2: Why must the levy measure have a second non-central moment about the origin? I've heard people saying that it's because a levy process has square-summable jump heights, but this isn't satisfactory from a measure-theoretic standpoint, nor does it really justify why the condition holds for IDDs in an abstract sense.
StackExchange has two good posts on this topic [2,3], but both make use of a random variable heuristic.
References:
- Sato, K. Levy Processes and Infinitely Divisible Distributions. Cambridge University Press, 1999. ISBN: 9780521553025
- https://math.stackexchange.com/a/4029367/527316
- https://math.stackexchange.com/a/890090/527316
