Abstract
This note deals with the orthogonality between sequences of random variables. The main idea of the note is to apply the results on equidistant systems of points in a Hilbert space to the case of the space L 2(Ω, F, ℙ) of real square integrable random variables. The main result gives a necessary and sufficient condition for a particular sequence of random variables (elements of which are taken from sets of equidistant elements of L 2(Ω, F, ℙ) to be orthogonal to some other sequence in L 2(Ω, F, ℙ). The result obtained is interesting from the point of view of the time series analysis, since it can be applied to a class of sequences random variables that exhibit a monotonically increasing variance. An application to ergodic theorem is also provided.
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References
E. M. E. Wermuth: A remark on equidistance in Hilbert spaces. Linear Algebra Appl. 236 (1996), 105–111. zbl
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The research of A. Volodin was partially supported by the National Science and Engineering Research Council of Canada.
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Triacca, U., Volodin, A. On a characterization of orthogonality with respect to particular sequences of random variables in L 2 . Appl Math 55, 329–335 (2010). https://doi.org/10.1007/s10492-010-0024-6
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DOI: https://doi.org/10.1007/s10492-010-0024-6