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Zap. Nauchn. Sem. POMI, 2015, Volume 441, Pages 239–262 (Mi znsl6236)  

This article is cited in 6 scientific papers (total in 6 papers)

A functional CLT for fields of commuting transformations via martingale approximation

Ch. Cunya, J. Dedeckerb, D. Volnýc

a Laboratoire MAS, Centrale-Supelec, Grande Voie des Vignes, 92295 Chatenay-Malabry cedex, France
b Laboratoire MAP5 (UMR 8145), Université Paris Descartes, Sorbonne Paris Cité, 45 rue des Saints Pères, 75270 Paris Cedex 06, France
c Laboratoire de Mathématiques Raphaël Salem (UMR 6085), Université de Rouen, Avenue de l'Universit, BP.12 76801 Saint-Etienne du Rouvray, France

Abstract: We consider a field $f\circ T^{i_1}_1\circ…\circ T_d^{i_d}$, where $T_1,…,T_d$ are completely commuting transformations in the sense of Gordin. If one of these transformations is ergodic, we give sufficient conditions in the spirit of Hannan under which the partial sum process indexed by quadrants converges in distribution to a Brownian sheet. The proof combines a martingale approximation approach with a recent CLT for martingale random fields due to Volný. We apply our results to completely commuting endomorphisms of the $m$-torus. In that case, the conditions can be expressed in terms of the $L^2$-modulus of continuity of $f$.

Key words and phrases: random fields, reverse martingales, endomorphisms of the torus.

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English version:
Journal of Mathematical Sciences (New York), 2016, 219:5, 765–781

Bibliographic databases:

UDC: 519.2
Received: 12.10.2015
Language:

Citation: Ch. Cuny, J. Dedecker, D. Volný, “A functional CLT for fields of commuting transformations via martingale approximation”, Probability and statistics. Part 22, Zap. Nauchn. Sem. POMI, 441, POMI, St. Petersburg, 2015, 239–262; J. Math. Sci. (N. Y.), 219:5 (2016), 765–781

Citation in format AMSBIB
\Bibitem{CunDedVol15}
\by Ch.~Cuny, J.~Dedecker, D.~Voln\'y
\paper A functional CLT for fields of commuting transformations via martingale approximation
\inbook Probability and statistics. Part~22
\serial Zap. Nauchn. Sem. POMI
\yr 2015
\vol 441
\pages 239--262
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl6236}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3504508}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2016
\vol 219
\issue 5
\pages 765--781
\crossref{https://doi.org/10.1007/s10958-016-3145-y}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. D. Volny, “Martingale-coboundary representation for stationary random fields”, Stoch. Dyn., 18:2 (2018), 1850011  crossref  mathscinet  zmath  isi  scopus
    2. P. Magda, N. Zhang, “Martingale approximations for random fields”, Electron. Commun. Probab., 23 (2018), 28  crossref  mathscinet  zmath  isi  scopus
    3. D. Giraudo, “Invariance principle via orthomartingale approximation”, Stoch. Dyn., 18:6 (2018), 1850043  crossref  mathscinet  zmath  isi  scopus
    4. M. Peligrad, N. Zhang, “On the normal approximation for random fields via martingale methods”, Stoch. Process. Their Appl., 128:4 (2018), 1333–1346  crossref  mathscinet  zmath  isi  scopus
    5. Peligrad M., Zhang N., “Central Limit Theorem For Fourier Transform and Periodogram of Random Fields”, Bernoulli, 25:1 (2019), 499–520  crossref  mathscinet  zmath  isi  scopus
    6. Volny D., “On Limit Theorems For Fields of Martingale Differences”, Stoch. Process. Their Appl., 129:3 (2019), 841–859  crossref  mathscinet  zmath  isi  scopus
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