The invariance principle for a class of dependent random fields
Author:
D. V. Poryvaĭ
Translated by:
V. Semenov
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 Free Access
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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 $s$, $s>2$, exist. The dependence structure of the field is described by the $\beta _1$-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.
References
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References
- A. V. Bulinskiĭ, Limit Theorems under Weak Dependence Conditions, Moscow University, Moscow, 1989. (Russian)
- A. V. Bulinskiĭ and E. Shabanovich, Asymptotic behavior of some functionals of positively or negatively dependent random fields, Fundam. Prikl. Mat. 4 (1998), no. 2, 479–492. (Russian) MR 1801168 (2001h:60033)
- 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)
- G. Bennett, Probability inequalities for the sum of independent random variables, J. Amer. Statist. Assoc. 57 (1962), 33–45.
- H. C. P. Berbee, Random Walks with Stationary Increments and Renewal Theory, Mathematical Centre Tracts, vol. 112, Amsterdam, 1979. MR 0547109 (81e:60093)
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- P. Doukhan, Mixing. Properties and Examples, Lecture Notes in Statistics, vol. 85, Springer, New York, 1994. MR 1312160 (96b:60090)
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- R. V. Fernique, Regularité des trajectoires des fonctions aléatoires gaussiennes, Lecture Notes in Math. 480, Springer, Berlin, 1975, pp. 1–96. MR 0413238 (54:1355)
- 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)
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Additional Information
D. V. Poryvaĭ
Affiliation:
Department of Probability Theory, Mechanics and Mathematics Faculty, Moscow State University, Moscow, Russia
Email:
denis@orc.ru
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
