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On regression representations of stochastic processes. (English) Zbl 0779.60058
Two types of so-called regression representations of a discrete time stochastic process \(X=(X_ n)_{n\in N}\) are developed. The one has the form \(X_ n=f_ n(X_ 1,\ldots,X_{n-1},U_ n)\), \(X_ 1=f_ 1(U_ 1)\), and is called Markov regression on \(X\) and the other is of the form \(X_ n=g_ n(U_ 1,\ldots,U_ n)\) and is called standard representation of \(X\) (on \((U_ n)_{n\in N})\). In both cases \((U_ n)_{n\in N}\) is assumed to be an i.i.d. sequence (the innovations) with \(U_ 1\) uniformly distributed on the interval \((0,1)\). Generalizing results of Ferguson (1967) and Skorokhod (1976) such representations are constructed for real valued processes. Under additional assumptions (e.g. \(X\) an \(m\)-dependent sequence or an \(m\)-Markov chain) the situation simplifies and so-called \((m+1)\)-block factor representations are derived. This gives reason to investigate the correspondence between both representations.
Reviewer: W.Schenk (Dresden)

MSC:
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
28D05 Measure-preserving transformations
60G10 Stationary stochastic processes
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