Hostname: page-component-5b777bbd6c-6lqsf Total loading time: 0 Render date: 2025-06-25T07:11:23.644Z Has data issue: false hasContentIssue false
  • English
  • Français

Exponential inequalities and functional central limit theorems for random fields

Published online by Cambridge University Press:  15 August 2002

Jérôme Dedecker
Jérôme Dedecker*
Affiliation:
LSTA, Université de Paris 6, 175 rue du Chevaleret, 75013 Paris Cedex 05, France; dedecker@ccr.jussieu.fr.
Get access

Abstract

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

Keywords

Functional central limit theoremstationary random fieldsmoment inequalitiesexponential inequalitiesmixingmetric entropychaining.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 5 , 2001 , pp. 77 - 104
Copyright
© EDP Sciences, SMAI, 2001

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

P. Hall and C.C. Heyde, Martingale Limit Theory and its Applications. Academic Press, New York (1980).
Alexander, K.S. and Pyke, R., A uniform central limit theorem for set-indexed partial-sum processes with finite variance. Ann. Probab. 14 (1986) 582-597. CrossRef
Azuma, K., Weighted sums of certain dependent random fields. Tôhoku Math. J. (2) 19 (1967) 357-367. CrossRef
Bass, R.F., Law of the iterated logarithm for set-indexed partial sum processes with finite variance. Z. Wahrsch. Verw. Gebiete. 70 (1985) 591-608. CrossRef
Basu, A.K. and Dorea, C.C.Y., On functional central limit theorem for stationary martingale random fields. Acta Math. Hungar. 33 (1979) 307-316. CrossRef
Bickel, P.J. and Wichura, M.J., Convergence criteria for multiparameter stochastic processes and some applications. Ann. Math. Statist. 42 (1971) 1656-1670. CrossRef
Bradley, R., A caution on mixing conditions for random fields. Statist. Probab. Lett. 8 (1989) 489-491. CrossRef
Chen, D., A uniform central limit theorem for nonuniform Φ-mixing random fields. Ann. Probab. 19 (1991) 636-649. CrossRef
Dedecker, J., A central limit theorem for stationary random fields. Probab. Theory Related Fields 110 (1998) 397-426. CrossRef
Dedecker, J. and Rio, E., On the functional central limit theorem for stationary processes. Ann. Inst. H. Poincaré Probab. Statist. 36 (2000) 1-34. CrossRef
Dobrushin, R.L., The description of a random field by means of conditional probabilities and conditions of its regularity. Theory Probab. Appl. 13 (1968) 197-224.
R.L. Dobrushin and S. Shlosman, constructive criterion for the uniqueness of Gibbs fields, Statistical physics and dynamical systems. Birkhauser (1985) 347-370.
P. Doukhan, Mixing: Properties and Examples. Springer, Berlin, Lecture Notes in Statist. 85 (1994).
Doukhan, P., León, J. and Portal, F., 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.
Dudley, R.M., Sample functions of the Gaussian process. Ann. Probab. 1 (1973) 66-103. CrossRef
Goldie, C.M. and Greenwood, P.E., Variance of set-indexed sums of mixing random variables and weak convergence of set-indexed processes. Ann. Probab. 14 (1986) 817-839. CrossRef
Goldie, C.M. and Morrow, G.J., Central limit questions for random fields, Dependence in probability and statistics. Progr. Probab. Statist. 11 (1986) 275-289.
Higuchi, Y., Coexistence of infinite (*)-clusters II. Ising percolation in two dimensions. Probab. Theory Related Fields 97 (1993) 1-33. CrossRef
Laroche, E., Hypercontractivité pour des systèmes de spins de portée infinie. Probab. Theory Related Fields 101 (1995) 89-132. CrossRef
M. Ledoux and M. Talagrand, Probability in Banach Spaces. Springer, New York (1991).
Lezaud, P., Chernoff-type bound for finite Markov chains. Ann. Appl. Probab. 8 (1998) 849-867.
Martinelli, F. and Olivieri, E., Approach to Equilibrium of Glauber Dynamics in the One Phase Region. I. The Attractive Case. Comm. Math. Phys. 161 (1994) 447-486. CrossRef
Peligrad, M., A note on two measures of dependence and mixing sequences. Adv. in Appl. Probab. 15 (1983) 461-464. CrossRef
G. Perera, Geometry of ${\mathbb Z}^d$ and the central limit theorem for weakly dependent random fields. J. Theoret. Probab. 10 (1997).
Pinelis, I.F., Optimum bounds for the distribution of martingales in Banach spaces. Ann. Probab. 22 (1994) 1679-1706. CrossRef
Rio, E., Covariance inequalities for strongly mixing processes. Ann. Inst. H. Poincaré 29 (1993) 587-597.
E. Rio, Théorèmes limites pour les suites de variables aléatoires faiblement dépendantes. Springer, Berlin, Collect. Math. Apll. 31 (2000).
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).
Schonmann, R.H. and Shlosman, S.B., Complete Analyticity for 2D Ising Completed. Comm. Math. Phys. 170 (1995) 453-482. CrossRef
Serfling, R.J., Contributions to Central Limit Theory For Dependent Variables. Ann. Math. Statist. 39 (1968) 1158-1175. CrossRef