The Annals of Probability

Hilbert Space Representations of $m$-Dependent Processes

Vincent De Valk

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A representation of one-dependent processes is given in terms of Hilbert spaces, vectors and bounded linear operators on Hilbert spaces. This generalizes a construction of one-dependent processes that are not two-block-factors. We show that all one-dependent processes admit a representation. We prove that if there is in the Hilbert space a closed convex cone that is invariant under certain operators and that is spanned by a finite number of linearly independent vectors, then the corresponding process is a two-block-factor of an independent process. Apparently the difference between two-block-factors and non-two-block-factors is determined by the geometry of invariant cones. The dimension of the smallest Hilbert space that represents a process is a measure for the complexity of the structure of the process. For two-valued one-dependent processes, if there is a cylinder with measure equal to zero, then this process can be represented by a Hilbert space with dimension smaller than or equal to the length of this cylinder. In the two-valued case a cylinder (with measure equal to zero) whose length is minimal and less than or equal to 7 is symmetric. We generalize the concept of Hilbert space representation to $m$-dependent processes and it turns out that all $m$-dependent processes admit a representation. Several theorems can be generalized to $m$-dependent processes.

Article information

Ann. Probab., Volume 21, Number 3 (1993), 1550-1570.

First available in Project Euclid: 19 April 2007

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Primary: 60G10: Stationary processes
Secondary: 28D05: Measure-preserving transformations 54H20: Topological dynamics [See also 28Dxx, 37Bxx] 52A20: Convex sets in n dimensions (including convex hypersurfaces) [See also 53A07, 53C45]

One-dependence block-factors Hilbert space representations stationary process $m$-dependence dynamical systems zero-cylinders invariant cones


Valk, Vincent De. Hilbert Space Representations of $m$-Dependent Processes. Ann. Probab. 21 (1993), no. 3, 1550--1570. doi:10.1214/aop/1176989130.

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