Uniqueness of a martingale-coboundary decomposition of stationary processes

Pavel Samek; Dalibor Volný

Commentationes Mathematicae Universitatis Carolinae (1992)

  • Volume: 33, Issue: 1, page 113-119
  • ISSN: 0010-2628

Abstract

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In the limit theory for strictly stationary processes f ∘ T i , i ∈ ℤ , the decomposition f = m + g - g ∘ T proved to be very useful; here T is a bimeasurable and measure preserving transformation an ( m ∘ T i ) is a martingale difference sequence. We shall study the uniqueness of the decomposition when the filtration of ( m ∘ 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]).

How to cite

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Samek, Pavel, and Volný, Dalibor. "Uniqueness of a martingale-coboundary decomposition of stationary processes." Commentationes Mathematicae Universitatis Carolinae 33.1 (1992): 113-119. <http://eudml.org/doc/247385>.

@article{Samek1992,
abstract = {In the limit theory for strictly stationary processes $f\circ T^i, i\in \mathbb \{Z\}$, 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]).},
author = {Samek, Pavel, Volný, Dalibor},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {strictly stationary process; approximating martingale; coboundary; approximating martingale; coboundary; strictly stationary processes; martingale difference sequence; decomposition},
language = {eng},
number = {1},
pages = {113-119},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Uniqueness of a martingale-coboundary decomposition of stationary processes},
url = {http://eudml.org/doc/247385},
volume = {33},
year = {1992},
}

TY - JOUR
AU - Samek, Pavel
AU - Volný, Dalibor
TI - Uniqueness of a martingale-coboundary decomposition of stationary processes
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1992
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 33
IS - 1
SP - 113
EP - 119
AB - In the limit theory for strictly stationary processes $f\circ T^i, i\in \mathbb {Z}$, 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]).
LA - eng
KW - strictly stationary process; approximating martingale; coboundary; approximating martingale; coboundary; strictly stationary processes; martingale difference sequence; decomposition
UR - http://eudml.org/doc/247385
ER -

References

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  7. Hall P., Heyde C.C., Martingal Limit Theory and its Application, Academic Press New York (1980). (1980) MR0624435
  8. Jacobs K., Lecture Notes on Ergodic Theory, Part I Matematisk Institut Aarhus Universitet Aarhus (1962-63). (1962-63) Zbl0196.31301
  9. Philipp W., Stout W., Almost Sure Invariance Principle for Partial Sums of Weakly Dependent Random Variables, Memoirs AMS 161 Providence, Rhode Island (1975). (1975) 
  10. Shiryaev A.N., Probability (in Russian), Nauka, Moscow, 1989. MR1024077
  11. Volný, D., Martingale decompositions of stationary processes, Yokoyama Math. J. 35 (1987), 113-121. (1987) MR0928378
  12. Volný, D., Approximating martingales and the central limit theorem for strictly stationary processes, to appear in Stoch. Processes and their Appl. MR1198662
  13. Volný, D., Martingale approximation of stationary processes: the choice of filtration, submitted for publication. 

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