# Questions tagged [levy-processes]

Theory and applications of Lévy processes (stochastic processes with stationary and independent increments): e.g. path properties, stochastic differential equations driven by jump-type processes, fluctuation theory of Lévy processes, queuing theory.

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### Show that $\int_{\mathbb R^p} x \nu(dx)=0$ in a question related to a certain convergence of measures

Consider the sequence of estochastic processes $(X_n, n \geq 1)$, where $X_n = (X_{t;n})_{t\in \mathbb Z}$ and: \begin{equation}\label{I}\tag{I} X_{t;n} = \sum_{j=0}^\infty \theta_{jn} \varepsilon_{t-...
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### Supremum process of a Cauchy RV

I've asked the same question on stats.stackexchange a week ago to no avail, so here we go again: Suppose $X_i$ are $\mathrm{Cauchy}(0,~\gamma)$ IID RV's. Does an expression exist for the CDF of the ...
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### Step in the derivation of the total idle time distribution of an M/G/1 queue

I'm trying to work my way through the proof of Thm. 1.11 in Kyprianou's Introductory Lectures on Fluctuations of Levy Processes with Applications but really struggle to understand the following step. ...
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### How to show that $\int x \,d\nu = 0$ using a pseudo-weak convergence of measures?

I have a sequence of $p$-dimensional infinitely divisible random vectors $S_n'$, such that $S_n' \Longrightarrow X$, as $n \to \infty$. Suppose the following assumptions The characteristic functions ...
1 vote
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### Interpretation of the Lévy measure of an infinitely divisible random vector

We know that a random vector $X$ is infinitely divisible (ID) if for all $n \in \mathbb N$, there exist $X_1^n,..., X_{n}^n$ i.i.d. random vectors such that: \begin{equation} X = X_1^n + ...+ X_n^...
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### If $(\exp(\mu_n))_{n\in\mathbb N}$ is weakly convergent, is the normalized sequence convergent as well?

Let $E$ be a metric space and $\mathcal M(E)$ denoote the space of finite signed measures on $\mathcal B(E)$ equipped with the total variation norm $\left\|\;\cdot\;\right\|$. I would like to know ...