Abstract
We exhibit a stationary countably-valued process {Vn} ∞−∞ which is deterministic, but which is nondeterministic in the sense that whenever ...n−2<n−1<n0<n1<... are indices with no two consecutive, then\(\{ V_{n_i } |i \in \mathbb{Z}\} \) is an independent process. This answers a question of Ref. 1.
In addition, althoughn↦Vn is deterministic, its time-reversaln↦V−n is not deterministic.
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References
King, J. L. (1992). Dilemma of the Sleeping Stockbroker,Amer. Math. Monthly 99, 4, 335–338.
Parry, W. (1981).Topics in Ergodic Theory, Cambridge University Press.
Petersen, K. (1983).Ergodic Theory, Cambridge University Press.
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Partly supported by grant DMS-9248571. Partly supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University, contract DAAL03-91-C-0027. 72604-3435@CompuServe.COM.
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Kalikow, S., King, J.L. A countably-valued sleeping stockbroker process. J Theor Probab 7, 703–708 (1994). https://doi.org/10.1007/BF02214366
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DOI: https://doi.org/10.1007/BF02214366