# Stochastic continuity and approximation

Studia Mathematica (1996)

- Volume: 121, Issue: 1, page 15-33
- ISSN: 0039-3223

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topBrown, Leon, and Schreiber, Bertram. "Stochastic continuity and approximation." Studia Mathematica 121.1 (1996): 15-33. <http://eudml.org/doc/216339>.

@article{Brown1996,

abstract = {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.},

author = {Brown, Leon, Schreiber, Bertram},

journal = {Studia Mathematica},

keywords = {stochastic processes which are continuous in probability; rational approximation; continuity in probability},

language = {eng},

number = {1},

pages = {15-33},

title = {Stochastic continuity and approximation},

url = {http://eudml.org/doc/216339},

volume = {121},

year = {1996},

}

TY - JOUR

AU - Brown, Leon

AU - Schreiber, Bertram

TI - Stochastic continuity and approximation

JO - Studia Mathematica

PY - 1996

VL - 121

IS - 1

SP - 15

EP - 33

AB - 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.

LA - eng

KW - stochastic processes which are continuous in probability; rational approximation; continuity in probability

UR - http://eudml.org/doc/216339

ER -

## References

top- [1] Andrus, G. F., and Nishiura, T., Stochastic approximation of random functions, Rend. Mat. (6) 13 (1980), 593-615. Zbl0486.60060
- [2] Arakelyan, N. V., Uniform approximation on closed sets by entire functions, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 1187-1206 (in Russian). Zbl0143.29602
- [3] Arens, R., Extension of functions on fully normal spaces, Pacific J. Math. 2 (1952), 11-22. Zbl0046.11801
- [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. Zbl0687.60001
- [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. Zbl0053.26802
- [8] Dugué, D., Traité de statistique théorique et appliquée: analyse aléatoire, algèbre aléatoire, Masson, Paris, 1958. Zbl0084.14305
- [9] Dugundji, J., An extension of Tietze's theorem, Pacific J. Math. 1 (1951), 353-367. Zbl0043.38105
- [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. Zbl0060.28808
- [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. Zbl0213.40401
- [15] Getoor, R. K., The Brownian escape process, Ann. Probab. 7 (1979), 864-867. Zbl0416.60086
- [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. Zbl0296.28003
- [18] Istrătescu, V. I., and Onicescu, O., Approximation theorems for random functions, Rend. Mat. (6) 8 (1975), 65-81. Zbl0308.60037
- [19] Kakutani, S., Simultaneous extension of continuous functions considered as a positive linear operation, Japan. J. Math. 17 (1940), 1-4. Zbl0023.39603
- [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). Zbl0059.05902
- [22] Michael, E., Some extension theorems for continuous functions, Pacific J. Math. 3 (1953), 789-806. Zbl0052.11502
- [23] Pełczyński, A., On simultaneous extension of continuous functions, Studia Math. 24 (1964), 285-304; Supplement: Studia Math. 25 (1964), 157-161. Zbl0145.16204
- [24] Pełczyński, Linear extensions, linear averagings, and their application to linear topological classification of spaces of continuous functions, Dissertationes Math. (Rozprawy Mat.) 58 (1968). Zbl0165.14603
- [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. Zbl0239.46048
- [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. Zbl0162.09702
- [30] Vitushkin, Analytic capacity of sets in problems of approximation theory, Russian Math. Surveys 22 (1967), 139-200. Zbl0164.37701
- [31] Zalcman, L., Analytic Capacity and Rational Approximation, Lecture Notes in Math. 50, Springer, Berlin, 1968. Zbl0171.03701

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