Hilbert Space Representations of m-Dependent Processes

Vincent de Valk

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    Abstract

    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.

    Original languageEnglish
    Pages (from-to)1550-1570
    Number of pages21
    JournalAnnals of probability
    Volume21
    Issue number3
    DOIs
    Publication statusPublished - Jul-1993

    Keywords

    • ONE-DEPENDENCE
    • BLOCK-FACTORS
    • HILBERT SPACE REPRESENTATIONS
    • STATIONARY PROCESS
    • M-DEPENDENCE
    • DYNAMICAL SYSTEMS
    • ZERO-CYLINDERS
    • INVARIANT COES
    • CENTRAL-LIMIT-THEOREM
    • ASYMPTOTIC EXPANSIONS
    • RANDOM-FIELDS
    • SEQUENCES
    • VARIABLES

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