Weak stability and generalized weak convolution for random vectors and stochastic processes
| Autor: | Jolanta K. Misiewicz |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2006 |
| Předmět: |
60A10
Mathematics::Number Theory Probability (math.PR) scale mixture Mathematics::Spectral Theory 60A10 60B05 60E05 60E07 60E10 (Primary) Nonlinear Sciences::Exactly Solvable and Integrable Systems 60E07 FOS: Mathematics 60B05 weakly stable distribution 60E05 60E10 Mathematics - Probability symmetric stable distribution |
| Zdroj: | Dee Denteneer, Frank den Hollander, Evgeny Verbitskiy, eds., Dynamics & Stochastics: Festschrift in honor of M. S. Keane (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2006), 109-118 |
| ISSN: | 0749-2170 |
| Popis: | A random vector ${\bf X}$ is weakly stable iff for all $a,b\in \mathbb{R}$ there exists a random variable $\Theta$ such that $a{\bf X}+b{\bf X}'\stackrel{d}{=}{\bf X}\Theta$. This is equivalent (see \cite{MOU}) with the condition that for all random variables $Q_1,Q_2$ there exists a random variable $\Theta$ such that $$ X Q_1 + X' Q_2 \stackrel{d}{=} X \Theta, $$ where ${\bf X},{\bf X}',Q_1,Q_2,\Theta$ are independent. In this paper we define generalized convolution of measures defined by the formula $$ L(Q_1) \oplus_{\mu} L(Q_2) = L(\Theta), $$ if the equation $(*)$ holds for ${\bf X},Q_1,Q_2,\Theta$ and $\mu ={\cal L}(\Theta)$. We study here basic properties of this convolution, basic properties of $\oplus_{\mu}$-infinitely divisible distributions, $\oplus_{\mu}$-stable distributions and give a series of examples. Comment: Published at http://dx.doi.org/10.1214/074921706000000149 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org) |
| Databáze: | OpenAIRE |
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