Abstract
It is known that a Gaussian stochastic process can be expanded in a functional series with random independent coefficients. In the case where the process is continuous in mean but there exists no modification of it with continuous simple functions, the series does not converge uniformly. In what cases does it converge pointwise? This question reduces to the well-studied problem of the boundedness of the sample functions. It is shown that the pointwise convergence of the expansion mentioned above is equivalent to the continuity of the sample functions of the process in a certain separable metric. Some other properties of Gaussian processes and measures are considered, ansd generalizations to the non-Gaussian case are given.
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Literature cited
K. Ito and M. Nisio, “On the oscillation functions of Gaussian processes,” Math. Scand.,22, No. 1, 209–233 (1968).
N. C. Jain and G. Kallianpur, “Oscillation function of a multiparameter Gaussian process,” Nagoya Math. J.,47, 15–28 (1972).
V. N. Sudakov, “A remark on the criterion of continuity of Gaussian sample functions,” Lecture Notes in Math.,330, 444–454 (1973).
B. S. Tsirel'son, “Some properties of lacunary series and Gaussian measures connected with uniform versions of the Egorov and Lusin properties,” Teor. Veroyatn. Ee Primen.,20, No. 3 (1975) (to appear).
Yu. K. Belyaev, “Local properties of sample functions of stationary Gaussian processes,” Teor. Veroyatn. Ee Primen.,5, No. 1, 128–131 (1960).
Yu. K. Belyaev, “Continuity and Hölder's condition for sample functions of stationary Gaussian processes,” in: Proc. Fourth Berkeley Symp. Math. Statist. Prob., Vol. 2 (1961), pp. 23–33.
D. M. Eaves, “Sample functions of Gaussian random homogeneous fields are either continuous or very irregular,” Ann. Math. Statist.,38, No. 5, 1579–1582 (1967).
A. M. Vershik and V. N. Sudakov, “Probability measures in infinite-dimensional spaces,” Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst.,12, 7–67 (1969).
K. Ito, “The canonical modification of stochastic processes,” J. Math. Soc. Jpn.,20, Nos. 1–2, 130–150 (1968).
R. M. Dudley, “The sizes of compact subsets of Hilbert space and continuity of Gaussian processes,” J. Funct. Anal.,1, No. 3, 290–330 (1967).
A. M. Vershik, “Axiomatics of measure theory in linear spaces,” Dokl. Akad. Nauk SSSR,178, No. 2, 278–281 (1968).
K. Fernique, “Intégrabilité des vecteurs gaussiens,” CR Acad. Sci., Paris,270A, No. 25, 1698–1699 (1970).
V. N. Sudakov and B. S. Tsirel'son, “Extremal properties of semispaces for spherically invariant measures,” Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst.,41, 14–24 (1974).
A. Grothendieck, “Produits tensoriels topologiques et espaces nucléaires,” Mem. Am. Math. Soc.,l6 (1955).
V. A. Rokhlin, “On the basic concepts of measure theory,” Matem. Sb.,25(67), No. 1, 107–150 (1949).
K. Kuratowski, Topology, Vol. 2, Moscow (1969).
I. A. Ibragimov and Yu. V. Linnik, Independent and Stationary Related Quantities, Moscow (1965).
B. S. Cirel'son, I. A. Ibragimov, and V. N. Sudakov, “Norms of Gaussian sample functions,” Lecture Notes in Math. (1976) (to appear).
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Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 55, pp. 35–63, 1976.
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Tsirel'son, B.S. A natural modification of a random process and its application to stochastic functional series and Gaussian measures. J Math Sci 16, 940–956 (1981). https://doi.org/10.1007/BF01676139
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DOI: https://doi.org/10.1007/BF01676139