Uncorrelatedness and orthogonality for vector-valued processes

Loeb, Peter; Osswald, Horst; Sun, Yeneng; Zhang, Zhixiang

doi:10.1090/S0002-9947-03-03450-0

Uncorrelatedness and orthogonality for vector-valued processes
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by Peter A. Loeb, Horst Osswald, Yeneng Sun and Zhixiang Zhang PDF

Trans. Amer. Math. Soc. 356 (2004), 3209-3225 Request permission

Abstract:

For a square integrable vector-valued process $f$ on the Loeb product space, it is shown that vector orthogonality is almost equivalent to componentwise scalar orthogonality. Various characterizations of almost sure uncorrelatedness for $f$ are presented. The process $f$ is also related to multilinear forms on the target Hilbert space. Finally, a general structure result for $f$ involving the biorthogonal representation for the conditional expectation of $f$ with respect to the usual product $\sigma$-algebra is presented.

References

Sergio Albeverio, Raphael Høegh-Krohn, Jens Erik Fenstad, and Tom Lindstrøm, Nonstandard methods in stochastic analysis and mathematical physics, Pure and Applied Mathematics, vol. 122, Academic Press, Inc., Orlando, FL, 1986. MR 859372
Robert M. Anderson, A non-standard representation for Brownian motion and Itô integration, Israel J. Math. 25 (1976), no. 1-2, 15–46. MR 464380, DOI 10.1007/BF02756559
J. Berger, H. Osswald, Y. N. Sun, J. L. Wu, On nonstandard product measure spaces, Illinois J. Math. 46(2002), 319–330.
Vivek S. Borkar, Probability theory, Universitext, Springer-Verlag, New York, 1995. An advanced course. MR 1367959, DOI 10.1007/978-1-4612-0791-7
Giuseppe Da Prato and Jerzy Zabczyk, Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1992. MR 1207136, DOI 10.1017/CBO9780511666223
J. Diestel and J. J. Uhl Jr., Vector measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis. MR 0453964, DOI 10.1090/surv/015
H. Jerome Keisler, Hyperfinite model theory, Logic Colloquium 76 (Oxford, 1976) Studies in Logic and Found. Math., Vol. 87, North-Holland, Amsterdam, 1977, pp. 5–110. MR 0491155
H. J. Keisler and Y. N. Sun, A metric on probabilities, and products of Loeb spaces, J. London Math. Soc., to appear.
Peter A. Loeb, Conversion from nonstandard to standard measure spaces and applications in probability theory, Trans. Amer. Math. Soc. 211 (1975), 113–122. MR 390154, DOI 10.1090/S0002-9947-1975-0390154-8
Peter A. Loeb, A functional approach to nonstandard measure theory, Conference in modern analysis and probability (New Haven, Conn., 1982) Contemp. Math., vol. 26, Amer. Math. Soc., Providence, RI, 1984, pp. 251–261. MR 737406, DOI 10.1090/conm/026/737406
Manfred Wolff and Peter A. Loeb (eds.), Nonstandard analysis for the working mathematician, Mathematics and its Applications, vol. 510, Kluwer Academic Publishers, Dordrecht, 2000. MR 1790871, DOI 10.1007/978-94-011-4168-0
H. Osswald, Malliavin calculus in abstract Wiener space using infinitesimals, Advances in Mathematics 176(2003), 1–37.
Yeneng Sun, A theory of hyperfinite processes: the complete removal of individual uncertainty via exact LLN, J. Math. Econom. 29 (1998), no. 4, 419–503. MR 1627287, DOI 10.1016/S0304-4068(97)00036-0
Yeneng Sun, The almost equivalence of pairwise and mutual independence and the duality with exchangeability, Probab. Theory Related Fields 112 (1998), no. 3, 425–456. MR 1660898, DOI 10.1007/s004400050196
Kôsaku Yosida, Functional analysis, 6th ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 123, Springer-Verlag, Berlin-New York, 1980. MR 617913