Metadata
Abstract
The now classical convergence in distribution theorem for well normalized sums of stationary martingale increments has been extended to multi-indexed martingale increments. In the present article we make progress in the identification of the limit law.
In dimension one, as soon as the stationary martingale increments form an ergodic process, the limit law is normal, and it is still the case for multi-indexed martingale increments when one of the processes defined by one coordinate of the multidimensional time is ergodic. In the general case, the limit may be non normal.
In the present paper we establish links between the dynamical properties of the $\mathbb{Z}^d$-measure preserving action associated to the stationary random field (like the positivity of the entropy of some factors) and the existence of a non normal limit law. The identification of a natural factor on which the $\mathbb{Z}^d$-action is of product type is a crucial step in this approach.
References
[Bil65] Ergodic theory and information, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, 1965, xiii+195 pages | MR | Zbl
[Bil68] Convergence of probability measures, John Wiley & Sons, 1968 | MR | Zbl
[CDV15] A functional CLT for fields of commuting transformations via martingale approximation, Zap. Nauchn. Semin. (POMI), Volume 441 (2015), pp. 239-262 | DOI | MR | Zbl
[Gir18] Invariance principle via orthomartingale approximation, Stoch. Dyn., Volume 18 (2018) no. 6, 1850043, 29 pages | DOI | MR | Zbl
[Gir24] Deviation inequality for Banach-valued orthomartingales, Stochastic Processes Appl., Volume 175 (2024), 104391, 20 pages | DOI | MR | Zbl
[Gla03] Ergodic theory via joinings, Mathematical Surveys and Monographs, 101, American Mathematical Society, 2003 | DOI | MR | Zbl
[Par69] Entropy and generators in ergodic theory, Mathematics Lecture Note Series, W. A. Benjamin, Inc., 1969, xii+124 pages | MR | Zbl
[Rio09] Moment inequalities for sums of dependent random variables under projective conditions, J. Theor. Probab., Volume 22 (2009) no. 1, pp. 146-163 | DOI | MR | Zbl
[Vol87] Martingale decompositions of stationary processes, Yokohama Math. J., Volume 35 (1987) no. 1-2, pp. 113-121 | MR | Zbl
[Vol15] A central limit theorem for fields of martingale differences, Comptes Rendus. Mathématique, Volume 353 (2015) no. 12, pp. 1159-1163 | DOI | MR | Zbl | Numdam
[Vol19] On limit theorems for fields of martingale differences, Stochastic Processes Appl., Volume 129 (2019) no. 3, pp. 841-859 | DOI | MR | Zbl
[WW13] A new condition for the invariance principle for stationary random fields, Stat. Sin., Volume 23 (2013) no. 4, pp. 1673-1696 | DOI | MR | Zbl