# Article

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Keywords:
strictly stationary process; approximating martingale; coboundary
Summary:
In the limit theory for strictly stationary processes $f\circ T^i, i\in\Bbb Z$, the decomposition $f=m+g-g\circ T$ proved to be very useful; here $T$ is a bimeasurable and measure preserving transformation an $(m\circ T^i)$ is a martingale difference sequence. We shall study the uniqueness of the decomposition when the filtration of $(m\circ T^i)$ is fixed. The case when the filtration varies is solved in [13]. The necessary and sufficient condition of the existence of the decomposition were given in [12] (for earlier and weaker versions of the results see [7]).
References:
[1] Bauer H.: Probability Theory and Elements of Measure Theory. Holt, Reinehart and Winston New York (1972). MR 0636091 | Zbl 0243.60004
[2] Billingsley P.: Ergodic Theory and Information. J. Wiley New York (1965). MR 0192027 | Zbl 0141.16702
[3] Cornfeld I.P., Fomin S.V., Sinai Ya.G.: Ergodic Theory. Springer-Verlag New York- Heidelberg-Berlin (1982). MR 0832433 | Zbl 0493.28007
[4] Eagleson G.K.: Martingale convergence to mixtures of infinitely divisible laws. Ann. Probab. 3 (1975), 557-562. MR 0378037 | Zbl 0319.60015
[5] Gilat D.: Some conditions under which two random variables are equal almost surely and simple proof of a theorem of Chung and Fuchs. Ann. Math. Statist. 42 (1971), 1647-1655. MR 0346898
[6] Gordin M.I.: The central limit theorem for stationary processes. Soviet Math. Dokl. 10 (1969), 1174-1176. MR 0251785 | Zbl 0212.50005
[7] Hall P., Heyde C.C.: Martingal Limit Theory and its Application. Academic Press New York (1980). MR 0624435
[8] Jacobs K.: Lecture Notes on Ergodic Theory. Part I Matematisk Institut Aarhus Universitet Aarhus (1962-63). Zbl 0196.31301
[9] Philipp W., Stout W.: Almost Sure Invariance Principle for Partial Sums of Weakly Dependent Random Variables. Memoirs AMS 161 Providence, Rhode Island (1975).
[10] Shiryaev A.N.: Probability (in Russian). Nauka, Moscow, 1989. MR 1024077
[11] Volný, D.: Martingale decompositions of stationary processes. Yokoyama Math. J. 35 (1987), 113-121. MR 0928378
[12] Volný, D.: Approximating martingales and the central limit theorem for strictly stationary processes. to appear in Stoch. Processes and their Appl. MR 1198662
[13] Volný, D.: Martingale approximation of stationary processes: the choice of filtration. submitted for publication.

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