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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
##### Keywords:
regression representations; factor representations
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##### References:
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