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 language | English |
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Pages (from-to) | 1550-1570 |
Number of pages | 21 |
Journal | Annals of probability |
Volume | 21 |
Issue number | 3 |
DOIs | |
Publication status | Published - 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