| Title:
|
Uniqueness of a martingale-coboundary decomposition of stationary processes (English) |
| Author:
|
Samek, Pavel |
| Author:
|
Volný, Dalibor |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
33 |
| Issue:
|
1 |
| Year:
|
1992 |
| Pages:
|
113-119 |
| . |
| Category:
|
math |
| . |
| 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]). (English) |
| Keyword:
|
strictly stationary process |
| Keyword:
|
approximating martingale |
| Keyword:
|
coboundary |
| MSC:
|
28D05 |
| MSC:
|
60G10 |
| idZBL:
|
Zbl 0753.60032 |
| idMR:
|
MR1173752 |
| . |
| Date available:
|
2009-01-08T17:53:57Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118476 |
| . |
| Reference:
|
[1] Bauer H.: Probability Theory and Elements of Measure Theory.Holt, Reinehart and Winston New York (1972). Zbl 0243.60004, MR 0636091 |
| Reference:
|
[2] Billingsley P.: Ergodic Theory and Information.J. Wiley New York (1965). Zbl 0141.16702, MR 0192027 |
| Reference:
|
[3] Cornfeld I.P., Fomin S.V., Sinai Ya.G.: Ergodic Theory.Springer-Verlag New York- Heidelberg-Berlin (1982). Zbl 0493.28007, MR 0832433 |
| Reference:
|
[4] Eagleson G.K.: Martingale convergence to mixtures of infinitely divisible laws.Ann. Probab. 3 (1975), 557-562. Zbl 0319.60015, MR 0378037 |
| Reference:
|
[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 |
| Reference:
|
[6] Gordin M.I.: The central limit theorem for stationary processes.Soviet Math. Dokl. 10 (1969), 1174-1176. Zbl 0212.50005, MR 0251785 |
| Reference:
|
[7] Hall P., Heyde C.C.: Martingal Limit Theory and its Application.Academic Press New York (1980). MR 0624435 |
| Reference:
|
[8] Jacobs K.: Lecture Notes on Ergodic Theory.Part I Matematisk Institut Aarhus Universitet Aarhus (1962-63). Zbl 0196.31301 |
| Reference:
|
[9] Philipp W., Stout W.: Almost Sure Invariance Principle for Partial Sums of Weakly Dependent Random Variables.Memoirs AMS 161 Providence, Rhode Island (1975). |
| Reference:
|
[10] Shiryaev A.N.: Probability (in Russian).Nauka, Moscow, 1989. MR 1024077 |
| Reference:
|
[11] Volný, D.: Martingale decompositions of stationary processes.Yokoyama Math. J. 35 (1987), 113-121. MR 0928378 |
| Reference:
|
[12] Volný, D.: Approximating martingales and the central limit theorem for strictly stationary processes.to appear in Stoch. Processes and their Appl. MR 1198662 |
| Reference:
|
[13] Volný, D.: Martingale approximation of stationary processes: the choice of filtration.submitted for publication. |
| . |
