Exponential inequalities and functional central limit theorems for random fields
ESAIM: Probability and Statistics, Tome 5 (2001) , pp. 77-104.

We establish new exponential inequalities for partial sums of random fields. Next, using classical chaining arguments, we give sufficient conditions for partial sum processes indexed by large classes of sets to converge to a set-indexed brownian motion. For stationary fields of bounded random variables, the condition is expressed in terms of a series of conditional expectations. For non-uniform φ-mixing random fields, we require both finite fourth moments and an algebraic decay of the mixing coefficients.

Classification : 37A50,  60F17
Mots clés : functional central limit theorem, stationary random fields, moment inequalities, exponential inequalities, mixing, metric entropy, chaining
@article{PS_2001__5__77_0,
     author = {Dedecker, J\'er\^ome},
     title = {Exponential inequalities and functional central limit theorems for random fields},
     journal = {ESAIM: Probability and Statistics},
     pages = {77--104},
     publisher = {EDP-Sciences},
     volume = {5},
     year = {2001},
     zbl = {1003.60033},
     mrnumber = {1875665},
     language = {en},
     url = {www.numdam.org/item/PS_2001__5__77_0/}
}
Dedecker, Jérôme. Exponential inequalities and functional central limit theorems for random fields. ESAIM: Probability and Statistics, Tome 5 (2001) , pp. 77-104. http://www.numdam.org/item/PS_2001__5__77_0/

[1] K.S. Alexander and R. Pyke, A uniform central limit theorem for set-indexed partial-sum processes with finite variance. Ann. Probab. 14 (1986) 582-597. | MR 832025 | Zbl 0595.60027

[2] K. Azuma, Weighted sums of certain dependent random fields. Tôhoku Math. J. (2) 19 (1967) 357-367. | MR 221571 | Zbl 0178.21103

[3] R.F. Bass, Law of the iterated logarithm for set-indexed partial sum processes with finite variance. Z. Wahrsch. Verw. Gebiete. 70 (1985) 591-608. | MR 807339 | Zbl 0575.60034

[4] A.K. Basu and C.C.Y. Dorea, On functional central limit theorem for stationary martingale random fields. Acta Math. Hungar. 33 (1979) 307-316. | MR 542479 | Zbl 0431.60037

[5] P.J. Bickel and M.J. Wichura, Convergence criteria for multiparameter stochastic processes and some applications. Ann. Math. Statist. 42 (1971) 1656-1670. | MR 383482 | Zbl 0265.60011

[6] R. Bradley, A caution on mixing conditions for random fields. Statist. Probab. Lett. 8 (1989) 489-491. | MR 1040812 | Zbl 0697.60054

[7] D. Chen, A uniform central limit theorem for nonuniform φ-mixing random fields. Ann. Probab. 19 (1991) 636-649. | MR 1106280 | Zbl 0735.60034

[8] J. Dedecker, A central limit theorem for stationary random fields. Probab. Theory Related Fields 110 (1998) 397-426. | MR 1616496 | Zbl 0902.60020

[9] J. Dedecker and E. Rio, On the functional central limit theorem for stationary processes. Ann. Inst. H. Poincaré Probab. Statist. 36 (2000) 1-34. | Numdam | MR 1743095 | Zbl 0949.60049

[10] R.L. Dobrushin, The description of a random field by means of conditional probabilities and conditions of its regularity. Theory Probab. Appl. 13 (1968) 197-224. | MR 231434 | Zbl 0184.40403

[11] R.L. Dobrushin and S. Shlosman, constructive criterion for the uniqueness of Gibbs fields, Statistical physics and dynamical systems. Birkhauser (1985) 347-370. | MR 821306 | Zbl 0569.46042

[12] P. Doukhan, Mixing: Properties and Examples. Springer, Berlin, Lecture Notes in Statist. 85 (1994). | MR 1312160 | Zbl 0801.60027

[13] P. Doukhan, J. León and F. Portal, Vitesse de convergence dans le théorème central limite pour des variables aléatoires mélangeantes à valeurs dans un espace de Hilbert. C. R. Acad. Sci. Paris Sér. I Math. 298 (1984) 305-308. | MR 765429 | Zbl 0557.60006

[14] R.M. Dudley, Sample functions of the Gaussian process. Ann. Probab. 1 (1973) 66-103. | MR 346884 | Zbl 0261.60033

[15] C.M. Goldie and P.E. Greenwood, Variance of set-indexed sums of mixing random variables and weak convergence of set-indexed processes. Ann. Probab. 14 (1986) 817-839. | MR 841586 | Zbl 0604.60032

[16] C.M. Goldie and G.J. Morrow, Central limit questions for random fields, Dependence in probability and statistics. Progr. Probab. Statist. 11 (1986) 275-289. | MR 991627 | Zbl 0605.60029

[17] P. Hall and C.C. Heyde, Martingale Limit Theory and its Applications. Academic Press, New York (1980). | MR 624435 | Zbl 0462.60045

[18] Y. Higuchi, Coexistence of infinite (*)-clusters II. Ising percolation in two dimensions. Probab. Theory Related Fields 97 (1993) 1-33. | MR 1240714 | Zbl 0794.60102

[19] E. Laroche, Hypercontractivité pour des systèmes de spins de portée infinie. Probab. Theory Related Fields 101 (1995) 89-132. | MR 1314176 | Zbl 0820.60082

[20] M. Ledoux and M. Talagrand, Probability in Banach Spaces. Springer, New York (1991). | MR 1102015 | Zbl 0748.60004

[21] P. Lezaud, Chernoff-type bound for finite Markov chains. Ann. Appl. Probab. 8 (1998) 849-867. | MR 1627795 | Zbl 0938.60027

[22] F. Martinelli and E. Olivieri, Approach to Equilibrium of Glauber Dynamics in the One Phase Region. I. The Attractive Case. Comm. Math. Phys. 161 (1994) 447-486. | MR 1269387 | Zbl 0793.60110

[23] M. Peligrad, A note on two measures of dependence and mixing sequences. Adv. in Appl. Probab. 15 (1983) 461-464. | MR 698829 | Zbl 0508.60033

[24] G. Perera, Geometry of d and the central limit theorem for weakly dependent random fields. J. Theoret. Probab. 10 (1997). | MR 1468394 | Zbl 0884.60022

[25] I.F. Pinelis, Optimum bounds for the distribution of martingales in Banach spaces. Ann. Probab. 22 (1994) 1679-1706. | MR 1331198 | Zbl 0836.60015

[26] E. Rio, Covariance inequalities for strongly mixing processes. Ann. Inst. H. Poincaré 29 (1993) 587-597. | EuDML 77471 | Numdam | MR 1251142 | Zbl 0798.60027

[27] E. Rio, Théorèmes limites pour les suites de variables aléatoires faiblement dépendantes. Springer, Berlin, Collect. Math. Apll. 31 (2000). | Zbl 0944.60008

[28] P.M. Samson, Inégalités de concentration de la mesure pour des chaînes de Markov et des processus φ-mélangeants, Thèse de doctorat de l'université Paul Sabatier (1998).

[29] R.H. Schonmann and S.B. Shlosman, Complete Analyticity for 2D Ising Completed. Comm. Math. Phys. 170 (1995) 453-482. | MR 1334405 | Zbl 0821.60097

[30] R.J. Serfling, Contributions to Central Limit Theory For Dependent Variables. Ann. Math. Statist. 39 (1968) 1158-1175. | MR 228053 | Zbl 0176.48004