3
votes
Accepted
Can we show that the characteristic function of an infinitely divisible probability measure has no zeros
$\newcommand\vpi\varphi\newcommand\R{\mathbb R}$
The approach involving (4) will not work, because Lévy's continuity theorem will guarantee that the pointwise limit of characteristic functions (c.f.'...
- 128k
2
votes
Accepted
Questions about Levy measure in the canonical representation of infinitely divisible distributions
I would discuss your questions partially under the following condition:
$$ \int_{-\infty}^\infty x^2\,dF(x)<\infty, \tag{1}$$
or equivalently $\varphi^{\prime\prime}(0)$ exists and finite or $\int_{...
- 724
2
votes
Accepted
Existence of unique convolution semigroups of probability measures on more general spaces then $\mathbb R^d$
A quick Google search on "infinitely divisible" and "Banach space" leads to Linde's Probability in Banach Spaces: Stable and Infinitely Divisible Distributions (John Wiley & ...
- 17.2k
1
vote
Accepted
If $\mu$ is an infinitely divisible probability measure on $[0,\infty)$, then the Lévy measure of $\mu$ is the vague limit of $n\mu^{*1/n}$
There are at least three ways to show that $n \mu^{*1/n}$ converges to $\nu$ vaguely in $\mathbb{R} \setminus \{0\}$. Let $X_t$ be the Lévy process such that $X_1$ has distribution $\mu$, and let $f$ ...
- 17.2k
1
vote
Accepted
How can we show this estimate for the convolution of two probability measures?
$\newcommand\ep\varepsilon\newcommand\de\delta$Let $\mu:=\mu_n$, $\nu:=\nu_n$, $B:=B_n$,
$$C_k:=\{x\in E:\mu(K_k-x)>1-\de_k\},$$
so that
$$B=\bigcap_k C_k.$$
We have
$$1-\ep_k<(\mu*\nu)(K_k)=\...
- 128k
1
vote
Characterization of the generator of a Lévy process using martingale problems
I think you can do it by computing the expectation of the Laplace transform of both the function f(X_t) and the integral (the one with the martingale vanishes) and then identity the Laplace transform ...
- 131
1
vote
Is this statement of the Lévy–Khintchine formula ill-posed?
Apologies if this is an inappropriate place to post a comment (I am a newbie). But I wanted to mention that an easier to read treatment of infinite divisibility is Chapter 3 of the classic book by ...
- 171
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