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On the filtering problem for stationary random Z2-fields

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3 Author(s)
Bulatek, W. ; Fac. of Math. & Comput. Sci., Nicholas Copernicus Univ., Torun, Poland ; Lemanczyk, M. ; Lesigne, E.

It is shown that whenever a stationary random field (Zn,m)n,m∈z is given by a Borel function f:Rz × Rz → R of two stationary processes (Xn)n∈z and (Ym)m∈z i.e., then (Zn, m) = (f((Xn+k)kεz, (Ym + ℓ )ℓ εz)) under a mild first coordinate univalence assumption on f, the process (Xn)n∈z is measurable with respect to (Zn,m)n,mεz whenever the process (Ym)m∈z is ergodic. The notion of universal filtering property of an ergodic stationary process is introduced, and then using ergodic theory methods it is shown that an ergodic stationary process has this property if and only if the centralizer of the dynamical system canonically associated with the process does not contain a nontrivial compact subgroup.

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Information Theory, IEEE Transactions on  (Volume:51 ,  Issue: 10 )