A central limit theorem for fields of martingale differences

Volný, Dalibor

Comptes Rendus. Mathématique, Tome 353 (2015) no. 12, pp. 1159-1163.

Le théorème limite centrale pour un champ aléatoire $f\circ T_{\underset{\_}{i}}$, $\underset{\_}{i}\in {\mathbb{Z}}^d$, de différences d'une martingale est démontré. Le résultat est connu pour les champs aléatoires de Bernoulli ; ici, l'ergodicité d'un seul générateur de l'action $T_{\underset{\_}{i}}$ est supposée.

We prove a central limit theorem for stationary random fields of martingale differences $f\circ T_{\underset{\_}{i}}$, $\underset{\_}{i}\in {\mathbb{Z}}^d$, where $T_{\underset{\_}{i}}$ is a ${\mathbb{Z}}^d$ action and the martingale is given by a commuting filtration. The result has been known for Bernoulli random fields; here only ergodicity of one of commuting transformations generating the ${\mathbb{Z}}^d$ action is supposed.

[1] Basu, A.K.; Dorea, C.C.Y. On functional central limit theorem for stationary martingale random fields, Acta Math. Acad. Sci. Hung., Volume 33 (1979) no. 3–4, pp. 307-316

[2] Biermé, H.; Durieu, O. Invariance principles for self-similar set-indexed random fields, Trans. Amer. Math. Soc., Volume 366 (2014), pp. 5963-5989

[3] Billingsley, P. On the Lindeberg–Lévy theorem for martingales, Proc. Amer. Math. Soc., Volume 12 (1961), pp. 788-792

[4] Cornfeld, I.P.; Fomin, S.V.; Sinai, Ya.G. Ergodic Theory, Springer-Verlag, Berlin, 1982

[5] El Machkouri, M.; Volný, D.; Wu, W.B. A central limit theorem for stationary random fields, Stoch. Process. Appl., Volume 123 (2013) no. 1, pp. 1-14

[6] Dedecker, J. A central limit theorem for stationary random fields, Probab. Theory Relat. Fields, Volume 110 (1998), pp. 397-426

[7] Gordin, M.I. The central limit theorem for stationary processes, Dokl. Akad. Nauk SSSR, Volume 188 (1969), pp. 739-741

[8] Gordin, M.I. Martingale-coboundary representation for a class of random fields, J. Math. Sci., Volume 163 (2009) no. 4, pp. 363-374 | DOI

[9] Hall, P.; Heyde, C. Martingale Limit Theory and Its Application, Academic Press, New York, 1980

[10] Ibragimov, I.A. A central limit theorem for a class of dependent random variables, Theory Probab. Appl., Volume 8 (1963), pp. 83-89

[11] Khosnevisan, D. Multiparameter Processes, an Introduction to Random Fields, Springer-Verlag, New York, 2002

[12] Klicnarová, J.; Volný, D.; Wang, Y. Limit theorems for weighted Bernoulli random fields under Hannan's condition (submitted for publication) | arXiv

[13] McLeish, D.L. Dependent central limit theorems and invariance principles, Ann. Probab., Volume 2 (1974), pp. 620-628

[14] Morkvenas, R. The invariance principle for martingales in the plane, Liet. Mat. Rink., Volume 24 (1984) no. 4, pp. 127-132

[15] Nahapetian, B. Billingsley–Ibragimov theorem for martingale-difference random fields and it applications to some models of classical statistical physics, C. R. Acad. Sci. Paris, Ser. I, Volume 320 (1995) no. 12, pp. 1539-1544

[16] Petersen, K. Ergodic Theory, Cambridge University Press, Cambridge, UK, 1990

[17] Poghosyan, S.; Roelly, S. Invariance principle for martingale-difference random fields, Stat. Probab. Lett., Volume 38 (1998) no. 3, pp. 235-245

[18] Volný, D.; Wang, Y. An invariance principle for stationary random fields under Hannan's condition, Stoch. Process. Appl., Volume 124 (2014), pp. 4012-4029

[19] D. Volný, in preparation.

[20] Wang, Y. An invariance principle for fractional Brownian sheets, J. Theor. Probab., Volume 27 (2014) no. 4, pp. 1124-1139

[21] Wang, Y.; Woodroofe, M. A new condition on invariance principles for stationary random fields, Stat. Sin., Volume 23 (2013) no. 4, pp. 1673-1696