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Harmonizability, $$V$$-boundedness, and stationary dilation of Banach- valued processes. (English) Zbl 0791.60025
Dudley, Richard M. (ed.) et al., Probability in Banach spaces, 8: Proceedings of the eighth international conference, held at Bowdoin College in summer of 1991. Boston, MA: Birkhäuser. Prog. Probab. 30, 189-205 (1992).
For Banach spaces $$X$$, $$Y$$ we consider the problem of factoring a family $$\{Z(\Delta):\Delta \in \Sigma\} \subseteq L(X,Y)$$, indexed by a $$\sigma$$-algebra, through a single Hilbert space $$H$$. We obtain a condition for such a factorization through a reproducing kernel Hilbert space involving an indexed family of positive operators taking values in the space of conjugate linear functionals on $$X$$. Under our condition, we get that the family $$\{Z(\Delta):\Delta \in \Sigma\}$$ has a spectral dilation in the sense that $$Z(\Delta)=SE(\Delta)R$$ where $$R:X \to H$$, $$S:H \to Y$$ are continuous linear operators and $$\{E(\Delta):\Delta \in \Sigma\}$$ is a countably additive spectral measure in $$H$$. As a consequence of this result we obtain necessary and sufficient conditions for the existence of an orthogonal dilation of a Banach space-valued measure $$\{Z(\Delta):\Delta \in \Sigma\}$$ of finite semivariation; $$Z(\Delta)=S\xi(\Delta)$$, $$\xi$$ an orthogonally scattered measure taking values in a Hilbert space $$H$$ and $$S:H \to Y$$ is continuous and linear taking values in the Banach space $$Y$$. As an application we obtain a representation of harmonizable stable processes in terms of stationary second order processes. Additionally, we propose definitions for harmonizable and $$V$$-bounded Banach space-valued processes indexed by a separable locally compact Abelian group. It has been brought to the attention of the author that our representation in Corollary 4.3 had been obtained by C. Houdré whose work was not referenced in this paper. The author welcomes the opportunity to correct this oversight.
For the entire collection see [Zbl 0773.00018].

##### MSC:
 60G10 Stationary stochastic processes 60B10 Convergence of probability measures