Weak stability and generalized weak convolution for random vectors and stochastic processes. (English) Zbl 1124.60003

Denteneer, Dee (ed.) et al., Dynamics and stochastics. Festschrift in honor of M. S. Keane. Selected papers based on the presentations at the conference ‘Dynamical systems, probability theory, and statistical mechanics’, Eindhoven, The Netherlands, January 3–7, 2005, on the occasion of the 65th birthday of Mike S. Keane. Beachwood, OH: IMS, Institute of Mathematical Statistics (ISBN 0-940600-64-1/pbk). Institute of Mathematical Statistics Lecture Notes - Monograph Series 48, 109-118 (2006).
Summary: A random vector X is weakly stable iff for all \(a,b\in \mathbb R\) there exists a random variable \(\Theta\) such that \(a{\mathbf X}+b{\mathbf X}'\overset{d}{=}{\mathbf X}\Theta\). This is equivalent [see J. K. Misiewicz, K. Oleszkiewicz and K. Urbanik, Stud. Math. 167, No. 3, 195–213 (2005; Zbl 1063.60017)] with the condition that for all random variables \(Q_1,Q_2\) there exists a random variable \(\Theta\) such that \[ {\mathbf X} Q_1 + {\mathbf X}' Q_2 \overset{d}{=} {\mathbf X} \Theta, \tag \(*\) \] where \({\mathbf X},{\mathbf X}',Q_1,Q_2,\Theta\) are independent. In this paper we define the generalized convolution of measures defined by the formula \( L(Q_1) \oplus_{\mu} L(Q_2) = L(\Theta)\), if the equation \((*)\) holds for \({\mathbf X},Q_1,Q_2,\Theta\) and \(\mu ={\mathcal L}(\Theta)\). 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.
For the entire collection see [Zbl 1113.60008].


60A10 Probabilistic measure theory
60B05 Probability measures on topological spaces
60E05 Probability distributions: general theory
60E07 Infinitely divisible distributions; stable distributions
60E10 Characteristic functions; other transforms


Zbl 1063.60017
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