Mathematics – Probability
Scientific paper
2006-08-09
IMS Lecture Notes--Monograph Series 2006, Vol. 48, 109-118
Mathematics
Probability
Published at http://dx.doi.org/10.1214/074921706000000149 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/
Scientific paper
10.1214/074921706000000149
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.
No associations
LandOfFree
Weak stability and generalized weak convolution for random vectors and stochastic processes does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Weak stability and generalized weak convolution for random vectors and stochastic processes, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Weak stability and generalized weak convolution for random vectors and stochastic processes will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-248133