Some Problems Related to Equivalence of Measures: Extension of Cylinder Set Measures and a Martingale Transformation.
NORTH CAROLINA UNIV CHAPEL HILL DEPT OF STATISTICS
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Let E be a linear one-to-one continuous map of the real and separable Hilbert space H into the real and separable Hilbert space K, with E having dense range. Gaussian cylinder set measures on H, defined by weak covariance operators, are considered. Such a cylinder set measure, and the map E, induces a Gaussian cylinder set measure on K. The first result is a characterization of the norm of the spaces K for which the induced measure extends to a probability measure on the Borel sets of K. This characterization is then used to study two probability measures on K, induced by E from two Gaussian cylinder set measures on H with known weak covariance operators. Conditions are obtained for the equivalence or orthogonality of the induced measures, and representations of the Radon-Nikodym derivative are given for the case when equivalence holds. Another problem considered is that of translating a continuous L sub 2 -bounded martingale, and making an associated absolutely continous substitution of measure. The problems considered in the report are related to the statistical theory of signal detection. Author
- Statistics and Probability