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Hilbert space representations of \(m\)-dependent processes. (English) Zbl 0802.60034
This paper deals with the investigation of the structure of \(m\)-dependent discrete time stochastic processes and can be viewed as continuation of the author’s successful research work in this field. The main result of this paper consists in showing that every stationary one-dependent random sequence (with finite state space) admits a so-called Hilbert space representation (HSR).
Further, if the Hilbert space which this representation is based on is two-dimensional, then the corresponding one-dependent sequence \(X_ n\), \(n\in N\), is a two-block-factor, i.e. there is a function \(f\) of two variables such that \(X_ n= f(Y_ n, Y_{n+1})\) with i.i.d. r.v.’s \(Y_ n\), \(n\in N\). As shown by J. Aaronson, D. Gilat and M. Keane [J. Theor. Probab. 5, No. 3, 545-561 (1992; Zbl 0754.60070)] there exist indeed examples of one-dependent sequences which cannot be a two-block factor.

MSC:
60G10 Stationary stochastic processes
60B11 Probability theory on linear topological spaces
60G05 Foundations of stochastic processes
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