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Bimeasures and nonstationary processes. (English) Zbl 0616.60009

Real and stochastic analysis, Wiley Ser. Probab. Math. Stat., Probab. Math. Stat., 7-118 (1986).
[For the entire collection see Zbl 0588.00021.]
This is a well-written paper over one hundred pages long. It deals with the interesting subject of bimeasures and their relation to second-order processes. The paper is divided into three parts. Part I consists of four sections which deal with the general theory of bimeasures and their integrals. The stochastic analysis is relegated to Part II which consists of four sections. Second-order process, in particular harmonizable ones, and their integral representation using bimeasures as well as the spectral domain of these processes are discussed in these sections.
Part III deals with applications to signal extraction, linear filtering, linear predictions theory for nonstationary processes and remarks on some related inference questions. This part consists of six sections.
The paper is mostly expository and embodies earlier results of the authors, although it contains some new developments. It is refreshing to see that many results scattered in literature are put together in this article. This will be of great help to newcomers to the field.
Reviewer’s remark: The reviewer was not able to completely understand the proof of Theorem 4 on page 76. The theorem deals with the completeness of the inner product space \({\mathcal L}^ 2(\beta)\) consisting of equivalence classes of matrix functions with respect to the bimeasure \(\beta\). (Consideration of the scalar case was not of much help.) The line of the proof is abstract and requires a tremendous amount of search through literature. This theorem is exciting if true in this generality, but I suspect trouble!
Reviewer: H.Salehi

MSC:

60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60B10 Convergence of probability measures
28B99 Set functions, measures and integrals with values in abstract spaces

Citations:

Zbl 0588.00021