Summary
If for a process \((\xi _n )_{n = - \infty }^\infty\) the conditional distribution of ξ n given the past does not depend on n for e.g. n≧0, then the process may be called a chain with infinite connections. Under a well-known continuity condition on this conditional distribution the process is shown to be distributed as an instantaneous function of a countable state Markov chain. Also under a certain weaker continuity condition uniqueness of the distributions of the stationary chains is obtained.
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Berbee, H. Chains with infinite connections: Uniqueness and Markov representation. Probab. Th. Rel. Fields 76, 243–253 (1987). https://doi.org/10.1007/BF00319986
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DOI: https://doi.org/10.1007/BF00319986