Stochastic continuity and approximation

• Autorzy: Leon Brown , Bertram M. Schreiber
• Języki publikacji: EN

Abstrakty

EN

This work is concerned with the study of stochastic processes which are continuous in probability, over various parameter spaces, from the point of view of approximation and extension. A stochastic version of the classical theorem of Mergelyan on polynomial approximation is shown to be valid for subsets of the plane whose boundaries are sets of rational approximation. In a similar vein, one can obtain a version in the context of continuity in probability of the theorem of Arakelyan on the uniform approximation of continuous functions on a closed set by entire functions. Locally bounded processes continuous in probability are characterized via operators from $L^1$-spaces to spaces of continuous functions. This characterization is utilized in a discussion of the problem of extension of the parameter space.

Kategorie tematyczne

• 30E10: Approximation in the complex domain
• 47B07: Operators defined by compactness properties
• 60G17: Sample path properties

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1996
• Tom: 121
• Numer: 1
• Strony: 15-33

Daty

wydano

1996

otrzymano

1995-07-06

poprawiono

1996-04-12

Twórcy

autor

Leon Brown

• Department of Mathematics, Wayne State University, Detroit, Michigan 48202, U.S.A.

autor

Bertram M. Schreiber

• Department of Mathematics, Wayne State University, Detroit, Michigan 48202, U.S.A.

Bibliografia

• [1] Andrus, G. F., and Nishiura, T., Stochastic approximation of random functions, Rend. Mat. (6) 13 (1980), 593-615.
• [2] Arakelyan, N. V., Uniform approximation on closed sets by entire functions, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 1187-1206 (in Russian).
• [3] Arens, R., Extension of functions on fully normal spaces, Pacific J. Math. 2 (1952), 11-22.
• [4] Blanc-Lapierre, A., and Fortet (transl. by J. Gani), R., Theory of Random Functions, Vol. 1, Gordon and Breach, New York, 1965.
• [5] Brown, L., and Schreiber, B. M., Approximation and extension of random functions, Monatsh. Math. 107 (1989), 111-123.
• [6] Diestel, J., and J. J. Uhl, Jr., Vector Measures, Math. Surveys 15, Amer. Math. Soc., Providence, R.I., 1977.
• [7] Doob, J. L., Stochastic Processes, Wiley, New York, 1953.
• [8] Dugué, D., Traité de statistique théorique et appliquée: analyse aléatoire, algèbre aléatoire, Masson, Paris, 1958.
• [9] Dugundji, J., An extension of Tietze's theorem, Pacific J. Math. 1 (1951), 353-367.
• [10] Dugundji, Topology, Allyn and Bacon, Boston, 1966.
• [11] Fan, K., Sur l'approximation et l'intégration des fonctions aléatoires, Bull. Soc. Math. France 72 (1944), 97-117.
• [12] Fedorchuk, V. V., The fundamentals of dimension theory, in: General Topology I, A. V. Arkhangel'skiĭ and L. S. Pontryagin (eds.), Encyclopaedia Math. Sci. 17, Springer, Berlin, 1990, 91-192.
• [13] Fernique, X., Les fonctions aléatoires cadlag, la compacité de leurs lois, Liet. Mat. Rink. 34 (1994), 288-306.
• [14] Gamelin, T. W., Uniform Algebras, Prentice-Hall, Englewood Cliffs, N.J., 1969.
• [15] Getoor, R. K., The Brownian escape process, Ann. Probab. 7 (1979), 864-867.
• [16] Gikhman, I. I., and Skorokhod, A. V., Introduction to the Theory of Random Processes, Saunders, Philadelphia, Penn., 1969.
• [17] Himmelberg, C. J., Measurable relations, Fund. Math. 87 (1975), 53-72.
• [18] Istrătescu, V. I., and Onicescu, O., Approximation theorems for random functions, Rend. Mat. (6) 8 (1975), 65-81.
• [19] Kakutani, S., Simultaneous extension of continuous functions considered as a positive linear operation, Japan. J. Math. 17 (1940), 1-4.
• [20] Kelley, J. L., General Topology, Van Nostrand, New York, 1955.
• [21] Mergelyan, S. N., Uniform approximations of functions of a complex variable, Uspekhi Mat. Nauk 7 (2 (48)) (1952), 31-122 (in Russian); English transl.: Amer. Math. Soc. Transl. 101 (1954).
• [22] Michael, E., Some extension theorems for continuous functions, Pacific J. Math. 3 (1953), 789-806.
• [23] Pełczyński, A., On simultaneous extension of continuous functions, Studia Math. 24 (1964), 285-304; Supplement: Studia Math. 25 (1964), 157-161.
• [24] Pełczyński, Linear extensions, linear averagings, and their application to linear topological classification of spaces of continuous functions, Dissertationes Math. (Rozprawy Mat.) 58 (1968).
• [25] Rudin, W., Real and Complex Analysis, 3rd ed., McGraw-Hill, New York, 1987.
• [26] Runge, C., Zur Theorie der eindeutigen analytischen Funktionen, Acta Math. 6 (1885), 229-244.
• [27] Semadeni, Z., Simultaneous Extensions and Projections in Spaces of Continuous Functions, Lecture Notes, Aarhus Univ., May 1965.
• [28] Syski, R., Stochastic processes, in: Encyclopedia Statist. Sci. 8, S. Kotz and N. L. Johnson (eds.), Wiley-Interscience, New York, 1988, 836-851.
• [29] Vitushkin, A. G., Conditions on a set which are necessary and sufficient in order that any continuous function, analytic at its interior points, admit uniform approximation by rational fractions, Soviet Math. Dokl. 7 (1966), 1622-1625.
• [30] Vitushkin, Analytic capacity of sets in problems of approximation theory, Russian Math. Surveys 22 (1967), 139-200.
• [31] Zalcman, L., Analytic Capacity and Rational Approximation, Lecture Notes in Math. 50, Springer, Berlin, 1968.