Trace measures of a positive definite bimeasure

Dominique Dehay and Raymond Moché

Journal of Multivariate Analysis, 1992, vol. 40, issue 1, 115-131

Abstract: If a positive definite bimeasure, M: - --> C, on any measurable space ([Omega], ) is extendable to a complex measure M', on the product [sigma]-algebra [circle times operator] , one can define its trace M'[Delta] on the diagonal [Delta] of [Omega] - [Omega]. If M is not extendable, the definition of M'[Delta] fails, but the need for defining some measures on [Delta] which would act as M'[Delta] remains, especially in the spectral analysis and the analysis of the stationarity of weakly harmonizable processes. In this paper, we fulfill this need with two types of measures concentrated on [Delta] or, more generally, on simple curves of [Omega] - [Omega]. The first ones, which have been called quasi-trace measures, are constructed under the assumption that the given positive definite bimeasure M is [sigma]-finite. They have the expected properties. The second ones, which are called pseudo-trace measures, are constructed, whatever the positive definite bimeasure M be, but they very often lead to new problems.

Keywords: positive; definite; bimeasures; trace; measures; diagonal; measures; weakly; harmonizable; processes; asymptotically; stationary; stochastic; processes (search for similar items in EconPapers)
Date: 1992
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