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Dilations of $ V$-bounded stochastic processes indexed by a locally compact group


Author: Kari Ylinen
Journal: Proc. Amer. Math. Soc. 90 (1984), 378-380
MSC: Primary 43A30; Secondary 60G12
MathSciNet review: 728352
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Abstract: It is proved that a stochastic process (i.e., a Hilbert space valued function) indexed by a locally compact group is $ V$-bounded (i.e., weakly harmonizable in an appropriate sense) if, and only if, it can be expressed as an orthogonal projection of a process whose covariance function $ R$ satisfies $ R(s,t) = \rho ({t^{ - 1}}s) + \rho (s{t^{ - 1}})$ for some continuous positive-definite function $ \rho $. The result generalizes a well-known theorem due to H. Niemi, and depends on the noncommutative Grothendieck type inequality of G. Pisier.


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  • [1] S. D. Chatterji, Orthogonally scattered dilation of Hilbert space valued set functions, Measure theory, Oberwolfach 1981 (Oberwolfach, 1981) Lecture Notes in Math., vol. 945, Springer, Berlin-New York, 1982, pp. 269–281. MR 675292
  • [2] Stanisław Goldstein and Ryszard Jajte, Second-order fields over 𝑊*-algebras, Bull. Acad. Polon. Sci. Sér. Sci. Math. 30 (1982), no. 5-6, 255–260 (English, with Russian summary). MR 673262
  • [3] Paul R. Halmos, Normal dilations and extensions of operators, Summa Brasil. Math. 2 (1950), 125–134. MR 0044036
  • [4] Hannu Niemi, On stationary dilations and the linear prediction of certain stochastic processes, Soc. Sci. Fenn. Comment. Phys.-Math. 45 (1975), no. 4, 111–130. MR 0410893
  • [5] Gilles Pisier, Grothendieck’s theorem for noncommutative 𝐶*-algebras, with an appendix on Grothendieck’s constants, J. Funct. Anal. 29 (1978), no. 3, 397–415. MR 512252, 10.1016/0022-1236(78)90038-1
  • [6] Kari Ylinen, Fourier transforms of noncommutative analogues of vector measures and bimeasures with applications to stochastic processes, Ann. Acad. Sci. Fenn. Ser. A I Math. 1 (1975), no. 2, 355–385. MR 0399755

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DOI: https://doi.org/10.1090/S0002-9939-1984-0728352-4
Article copyright: © Copyright 1984 American Mathematical Society