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 -mixing random fields, we require both finite fourth moments and an algebraic decay of the mixing coefficients.
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}, mrnumber = {1875665}, zbl = {1003.60033}, language = {en}, url = {http://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] A uniform central limit theorem for set-indexed partial-sum processes with finite variance. Ann. Probab. 14 (1986) 582-597. | MR | Zbl
and ,[2] Weighted sums of certain dependent random fields. Tôhoku Math. J. (2) 19 (1967) 357-367. | MR | Zbl
,[3] Law of the iterated logarithm for set-indexed partial sum processes with finite variance. Z. Wahrsch. Verw. Gebiete. 70 (1985) 591-608. | MR | Zbl
,[4] On functional central limit theorem for stationary martingale random fields. Acta Math. Hungar. 33 (1979) 307-316. | MR | Zbl
and ,[5] Convergence criteria for multiparameter stochastic processes and some applications. Ann. Math. Statist. 42 (1971) 1656-1670. | MR | Zbl
and ,[6] A caution on mixing conditions for random fields. Statist. Probab. Lett. 8 (1989) 489-491. | MR | Zbl
,[7] A uniform central limit theorem for nonuniform -mixing random fields. Ann. Probab. 19 (1991) 636-649. | MR | Zbl
,[8] A central limit theorem for stationary random fields. Probab. Theory Related Fields 110 (1998) 397-426. | MR | Zbl
,[9] On the functional central limit theorem for stationary processes. Ann. Inst. H. Poincaré Probab. Statist. 36 (2000) 1-34. | Numdam | MR | Zbl
and ,[10] The description of a random field by means of conditional probabilities and conditions of its regularity. Theory Probab. Appl. 13 (1968) 197-224. | MR | Zbl
, and , constructive criterion for the uniqueness of Gibbs fields, Statistical physics and dynamical systems. Birkhauser (1985) 347-370. |[12] Mixing: Properties and Examples. Springer, Berlin, Lecture Notes in Statist. 85 (1994). | MR | Zbl
,[13] 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 | Zbl
, and ,[14] Sample functions of the Gaussian process. Ann. Probab. 1 (1973) 66-103. | MR | Zbl
,[15] Variance of set-indexed sums of mixing random variables and weak convergence of set-indexed processes. Ann. Probab. 14 (1986) 817-839. | MR | Zbl
and ,[16] Central limit questions for random fields, Dependence in probability and statistics. Progr. Probab. Statist. 11 (1986) 275-289. | MR | Zbl
and ,[17] Martingale Limit Theory and its Applications. Academic Press, New York (1980). | MR | Zbl
and ,[18] Coexistence of infinite ()-clusters II. Ising percolation in two dimensions. Probab. Theory Related Fields 97 (1993) 1-33. | MR | Zbl
,[19] Hypercontractivité pour des systèmes de spins de portée infinie. Probab. Theory Related Fields 101 (1995) 89-132. | MR | Zbl
,[20] Probability in Banach Spaces. Springer, New York (1991). | MR | Zbl
and ,[21] Chernoff-type bound for finite Markov chains. Ann. Appl. Probab. 8 (1998) 849-867. | MR | Zbl
,[22] Approach to Equilibrium of Glauber Dynamics in the One Phase Region. I. The Attractive Case. Comm. Math. Phys. 161 (1994) 447-486. | MR | Zbl
and ,[23] A note on two measures of dependence and mixing sequences. Adv. in Appl. Probab. 15 (1983) 461-464. | MR | Zbl
,[24] Geometry of and the central limit theorem for weakly dependent random fields. J. Theoret. Probab. 10 (1997). | MR | Zbl
,[25] Optimum bounds for the distribution of martingales in Banach spaces. Ann. Probab. 22 (1994) 1679-1706. | MR | Zbl
,[26] Covariance inequalities for strongly mixing processes. Ann. Inst. H. Poincaré 29 (1993) 587-597. | EuDML | Numdam | MR | Zbl
,[27] Théorèmes limites pour les suites de variables aléatoires faiblement dépendantes. Springer, Berlin, Collect. Math. Apll. 31 (2000). | Zbl
,[28] Inégalités de concentration de la mesure pour des chaînes de Markov et des processus -mélangeants, Thèse de doctorat de l'université Paul Sabatier (1998).
,[29] Complete Analyticity for 2D Ising Completed. Comm. Math. Phys. 170 (1995) 453-482. | MR | Zbl
and ,[30] Contributions to Central Limit Theory For Dependent Variables. Ann. Math. Statist. 39 (1968) 1158-1175. | MR | Zbl
,