Questions tagged [levy-processes]

Question related to Lévy processes, i.e. stochastically continuous processes with independent, stationary increments.

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Levy Process is a Feller Processs

I want to show that every Levy Process is a Feller Process. Let $X=(X_t)_{t\geq 0}$ be a Levy Process and $\mu_t(dy):=P\circ X^{-1}(dy)$ . I found a proof where it is shown that the transition ...
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What's so special about the Levy distribution?

The distribution with the PDF below is a particular case of the Levy distribution. $$f(x) = \sqrt{\frac{1}{\pi}}.\frac{e^{-\frac{1}{x}}}{x^{1.5}}$$ Someone said it was their favorite distribution. ...
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Are all discontinuous levy processes sparse?

As described by the title, I would like to know if all discontinuous Levy processes (or jump Levy process, such as Poisson Process, Cauchy process, Gamma Process, etc.) are sparse processes? The ...
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A problem related to the convergence of infinitely divisible distributions

We have a classic theorem about convergence of infinitely divisible (ID) distributions. Before, remember that we say that the ID random vector $X$ has a Lévy-Khintchine representation, for a suitable ...
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If $\int_{\mathbb{R}} \min(1,x^2)\nu(dx)<\infty$ then $\nu(A)< \infty$ for all A if $0$ is in the interior of $A^c$ [closed]

I am currently reading "Fluctuations of Lévy Processes with Applications" by Kyprianou, where he mentions (p.36 at the top) that if $\nu$ is a measure on $\mathbb{R}\setminus\{0\}$ ...
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Show that a Levy measure $\nu$ (which arises from a convergence of Infinitely Divisible random vectors) is such that $\int x d\nu(x)=0$

Let $(X_{jn})_{1\leq j \leq n}$ be a triangular array of $p-$dimensional random vectors (row independent). Suppose $X_{jn} \sim \mu_{jn}$ and  1. $\,\, E X_{jn}= \int_{\mathbb R^p} x d \mu_{jn}=0$   2....
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Show that a Levy measure (which arises from a convergence of Infinitely Divisible random vectors) has infinite total mass

Given an array of probability measures $(\mu_{jn})_{1\leq j \leq n}$, all defined on borelians of $\mathbb R^p$. Let $S_n'$ a sequence of $p-$dimensional random vectors with $S_n' \Longrightarrow X$ ...
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I'm working with the isotropic $\alpha$-stable Lévy process in $\mathbb{R}^d$ $(\alpha \in (0,1) \text{ and } d \geq 2)$. I know that the distribution of the first hitting of this process into the ...
If $\nu$ is a Levy measure, how to show that $\nu( (-\epsilon,\epsilon)^c ) <\infty$
We know that the Levy measure on borelians is a measure $\nu$ such that $$\int \frac{|x|^2}{1+ |x|^2} \nu (dx) < \infty$$ We can show that this is equivalent to: \...