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Series representations and Karhunen processes

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  • Kakihara, Yûichirô

Abstract

Three types of series representations are considered for Hilbert space valued second-order stochastic processes over the real line. These are called ordinary, modular, and tensor series representations. Relations among these representations are given in connection with the separability of the vector or modular time domain of a process. It is shown that, in a modular and a tensor series representation, the time-dependent deterministic functions form a modular basis and an orthonormal set in appropriate reproducing kernel spaces, respectively. Finally, Karhunen processes are defined in the Hilbert space valued case and their characterization is given. Namely, Karhunen processes are those which have separable modular time domains, or equivalently, those which have modular or tensor series representations.

Suggested Citation

  • Kakihara, Yûichirô, 1988. "Series representations and Karhunen processes," Journal of Multivariate Analysis, Elsevier, vol. 25(1), pages 10-24, April.
    • Handle: RePEc:eee:jmvana:v:25:y:1988:i:1:p:10-24
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