The invariance principle for a class of dependent random fields

Author:
D. V. Poryvai

Translated by:
V. Semenov

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **70** (2004).

Journal:
Theor. Probability and Math. Statist. **70** (2005), 123-134

MSC (2000):
Primary 60F17, 60G60

DOI:
https://doi.org/10.1090/S0094-9000-05-00636-8

Published electronically:
August 12, 2005

MathSciNet review:
2109829

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Sufficient conditions for the tightness of a family of distributions of partial sum set-indexed processes constructed from symmetric random fields are obtained in this paper. We require that the moments of order , , exist. The dependence structure of the field is described by the -mixing coefficients decreasing with a power rate. Assuming that a field is stationary and applying a result of D. Chen (1991) on the convergence of finite-dimensional distributions of the processes we obtain the invariance principle.

**1.**A. V. Bulinski,*Limit Theorems under Weak Dependence Conditions*, Moscow University, Moscow, 1989. (Russian)**2.**A. V. Bulinskiĭ and È. Shabanovich,*Asymptotic behavior of some functionals of positively and negatively dependent random fields*, Fundam. Prikl. Mat.**4**(1998), no. 2, 479–492 (Russian, with English and Russian summaries). MR**1801168****3.**K. Alexander and R. Pyke,*A uniform central limit theorem for set-indexed partial-sum processes with finite variance*, Ann. Probab.**14**(1986), no. 2, 582-597. MR**0832025 (88b:60084)****4.**G. Bennett,*Probability inequalities for the sum of independent random variables*, J. Amer. Statist. Assoc.**57**(1962), 33-45.**5.**H. C. P. Berbee,*Random Walks with Stationary Increments and Renewal Theory*, Mathematical Centre Tracts, vol. 112, Amsterdam, 1979. MR**0547109 (81e:60093)****6.**Dong Ching Chen,*A uniform central limit theorem for nonuniform 𝜙-mixing random fields*, Ann. Probab.**19**(1991), no. 2, 636–649. MR**1106280****7.**Jérôme Dedecker,*Exponential inequalities and functional central limit theorems for a random fields*, ESAIM Probab. Statist.**5**(2001), 77–104. MR**1875665**, https://doi.org/10.1051/ps:2001103**8.**Paul Doukhan,*Mixing*, Lecture Notes in Statistics, vol. 85, Springer-Verlag, New York, 1994. Properties and examples. MR**1312160****9.**R. M. Dudley,*Sample functions of the Gaussian process*, Ann. Probability**1**(1973), no. 1, 66–103. MR**0346884****10.**X. Fernique,*Regularité des trajectoires des fonctions aléatoires gaussiennes*, École d’Été de Probabilités de Saint-Flour, IV-1974, Springer, Berlin, 1975, pp. 1–96. Lecture Notes in Math., Vol. 480 (French). MR**0413238****11.**C. M. Goldie and P. Greenwood,*Variance of set-indexed sums of mixing random variables and weak convergence of set-indexed processes*, Ann. Probab.**14**(1985), 815-839. MR**0841586 (88e:60038b)****12.**Michel Ledoux and Michel Talagrand,*Probability in Banach spaces*, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 23, Springer-Verlag, Berlin, 1991. Isoperimetry and processes. MR**1102015**

Retrieve articles in *Theory of Probability and Mathematical Statistics*
with MSC (2000):
60F17,
60G60

Retrieve articles in all journals with MSC (2000): 60F17, 60G60

Additional Information

**D. V. Poryvai**

Affiliation:
Department of Probability Theory, Mechanics and Mathematics Faculty, Moscow State University, Moscow, Russia

Email:
denis@orc.ru

DOI:
https://doi.org/10.1090/S0094-9000-05-00636-8

Received by editor(s):
February 27, 2003

Published electronically:
August 12, 2005

Additional Notes:
Supported in part by the RFFI grant 03-01-00724.

Article copyright:
© Copyright 2005
American Mathematical Society