Abstract
For a Wiener field with an arbitrary finite number of parameters, we construct the law of the iterated logarithm in the functional form. We consider the problem for random fields of a certain type to reside within curvilinear boundaries without assuming that the Cairoli—Walsh condition is satisfied.
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References
P. Imkeller, “Stochastic calculus for continuous N-parameter strong martingales,” Stoch. Processes Their Applications, 20, 1–40 (1985).
B. V. Bondarev and G. G. Zhirnyi, “Some properties of multiparameter random fields,” Ukr. Mat. Zh., 47, No. 12, 1609–1622 (1995).
I. I. Gikhman, “Two-parameter martingales,” Usp. Mat. Nauk, 37, No. 6 (228), 1–28 (1982).
I. Fazekas, “On convergence of multiparameter strong martingales in Banach lattices,” Analysis Math., 10, 202–207 (1984).
B. V. Bondarev and G. G. Jirny, “On the functional of action for the multiparameter random fields,” Rand. Oper. Stoch. Equats., 3, No. 2, 37–46 (1995).
V. Strassen, “An invariance principle for the law of the iterated logarithm,” Z. Wahrsheinlichkeitstheorie verw. Gebiete, 3, No. 3, 211–226 (1964).
A. V. Bulinskii, “A new version of the functional law of iterated logarithm,” Teor. Ver. Primen., 25, No. 3, 502–512 (1980).
A. V. Bulinskii, Limit Theorems for Random Processes and Fields [in Russian], Moscow University, Moscow (1981).
L. L. Ponomarenko, “Stochastic integrals for multiparameter Brown motion and related stochastic equations,” Teor. Ver. Mat. Statist., 7, 100–109 (1972).
A. V. Bitsadze, Some Classes of Partial Differential Equations [in Russian], Nauka, Moscow (1981).
Additional information
Donetsk University, Donetsk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 7, pp. 883–894, July, 1997.
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Bondarev, B.V., Zhirnyi, G.G. Functional law of the iterated logarithm for fields and its applications. Ukr Math J 49, 989–1002 (1997). https://doi.org/10.1007/BF02528744
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DOI: https://doi.org/10.1007/BF02528744