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Decomposition of self-similar stable mixed moving averages. (English) Zbl 1007.60026
A standard Borel space (SBS) is a measurable space measurable isomorphic to a Borel subset of a complete separable metric space. When a SBS is equipped with a $$\sigma$$-finite measure $$\mu$$, it is called standard Lebesgue space (SLS). Let $$(X, \mathcal X, \mu)$$ be a SLS. A family $$\{\varphi_t\}_{t\in\mathbb R}$$ of measurable maps from $$X$$ onto $$X$$ form an additive flow if $$\varphi_0(x)=x$$ and $$\varphi_{t_1}(\varphi_{t_2}(x))=\varphi_{t_1+t_2}(x)$$. The flow $$\{\varphi_t\}$$ is nonsingular if $$\mu(\varphi_t^{-1}(B))=0$$ iff $$\mu(B)=0$$ for every $$t\in\mathbb R$$ and $$B\in\mathcal X$$. A measurable map $$a_t(x):\mathbb R\times X\mapsto \{-1,1\}$$ is a cocycle if $$a_{t_1+t_2}(x)=a_{t_2}(x)a_{t_1}(\varphi_{t_2}(x))$$. The authors investigate a symmetric $$\alpha$$-stable (S$$\alpha$$S) mixed moving average process $$X_{\alpha}$$ with stationary increments defined by $X_{\alpha}(t)=\int_X\int_{\mathbb R} (G(x, t+u) -G(x,u))M_{\alpha}(dx,du)$ where $$G$$ is a measurable function and $$M_{\alpha}$$ is a S$$\alpha$$S random measure with the control measure $$m_{\alpha}(dx,du)=\mu(dx)du$$. It is proved that a self-similar process $$X_{\alpha}$$ is determined by a nonsingular flow, a related cocycle and a semi-additive functional. A unique decomposition into two independent components $$X_{\alpha}\overset\text{d}= X_{\alpha}^D +X_{\alpha}^C$$ is established, where $$X_{\alpha}^D$$ is determined by a nonsingular dissipative flow and $$X_{\alpha}^C$$ by a nonsingular conservative flow.

##### MSC:
 60G18 Self-similar stochastic processes 60G52 Stable stochastic processes 60G57 Random measures
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