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Orthogonal expansions of vectors in a Hilbert space for non-Gaussian measures


Author: Yoshiaki Okazaki
Journal: Proc. Amer. Math. Soc. 86 (1982), 638-640
MSC: Primary 60B11
DOI: https://doi.org/10.1090/S0002-9939-1982-0674096-5
MathSciNet review: 674096
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Abstract: Let $ \mathcal{H}$ be a separable Hilbert space and $ \mu $ a probability Radon measure on $ \mathcal{H}$ of second order. Then there exist $ ({a_n}) \in {l^2}$, an O.N.S. $ ({x_n}) \subset \mathcal{H}$ and an O.N.S. $ ({\xi _n}) \subset H$ such that the orthogonal series $ \sum\nolimits_{n = 1}^\infty {{a_n}{\xi _n}(x){x_n}} $ converges in $ \mathcal{H}$ $ \mu $-almost everywhere and it holds that $ x = \sum\nolimits_{n = 1}^\infty {{a_n}{\xi _n}(x){x_n}} $, $ \mu $-almost everywhere, where $ H$ is the generating Hubert space of $ \mu $. In the case where $ \mu $ is a Gaussian measure, a similar result was proved by Kuelbs [2] in general Banach spaces.


References [Enhancements On Off] (What's this?)

  • [1] K. Ito and M. Nisio, On the convergence of sums of independent Banach space valued random variables, Osaka J. Math. 5 (1968), 35-48. MR 0235593 (38:3897)
  • [2] J. Kuelbs, Expansions of vectors in a Banach space related to Gaussian measures, Proc. Amer. Math. Soc. 27 (1971), 364-371. MR 0267615 (42:2517)
  • [3] -, A strong convergence theorem for Banach space valued random variables, Ann. Probab. 4 (1976), 744-771. MR 0420771 (54:8783)

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DOI: https://doi.org/10.1090/S0002-9939-1982-0674096-5
Keywords: Orthogonal expansion, generating Hilbert space
Article copyright: © Copyright 1982 American Mathematical Society

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