Norm convergent expansions for Gaussian processes in Banach spaces.
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- by Naresh C. Jain and G. Kallianpur
- Proc. Amer. Math. Soc. 25 (1970), 890-895
- DOI: https://doi.org/10.1090/S0002-9939-1970-0266304-1
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Abstract:
Several authors have recently shown that Brownian motion with continuous paths on $[0,1]$ can be expanded into a uniformly convergent (a.s.) orthogonal series in terms of a given complete orthonormal system (CONS) in its reproducing kernel Hilbert space (RKHS). In an earlier paper we generalized this result to Gaussian processes with continuous paths. Here we obtain such expansions for a Gaussian random variable taking values in an arbitrary separable Banach space. A related problem is also considered in which starting from a separable Hilbert space $H$ with a measurable norm $|| \cdot |{|_1}$ defined on it, it is shown that the corresponding abstract Wiener process has a $||\cdot |{|_1}$-convergent orthogonal expansion in terms of a CONS chosen from $H$.References
- S. Banach, Théorie des opérations linéaires, Monogr. Mat., PWN, Warsaw, 1932; reprint, Chelsea, New York, 1955. MR 17, 175.
- Leonard Gross, Measurable functions on Hilbert space, Trans. Amer. Math. Soc. 105 (1962), 372–390. MR 147606, DOI 10.1090/S0002-9947-1962-0147606-6
- Leonard Gross, Abstract Wiener spaces, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66) Univ. California Press, Berkeley, Calif., 1967, pp. 31–42. MR 0212152
- Kiyosi Itô and Makiko Nisio, On the convergence of sums of independent Banach space valued random variables, Osaka Math. J. 5 (1968), 35–48. MR 235593
- Naresh C. Jain and G. Kallianpur, A note on uniform convergence of stochastic processes, Ann. Math. Statist. 41 (1970), 1360–1362. MR 272050, DOI 10.1214/aoms/1177696914
- G. Kallianpur, Abstract Wiener processes and their reproducing kernel Hilbert spaces, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 17 (1971), 113–123. MR 281242, DOI 10.1007/BF00538863
Bibliographic Information
- © Copyright 1970 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 25 (1970), 890-895
- MSC: Primary 60.50
- DOI: https://doi.org/10.1090/S0002-9939-1970-0266304-1
- MathSciNet review: 0266304