Equivalent Gaussian Measures Whose R-N Derivative is the Exponential of a Diagonal Form.
NORTH CAROLINA UNIV CHAPEL HILL DEPT OF STATISTICS
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A simple necessary and sufficient condition, on a trace-class kernel K, is given in order for the existence of a measurable relative to the complete product sigma-algebra Gaussian process with covariance K. The integral of the exponential of a certain function of a Gaussian process with respect to the corresponding probability measure is calculated, explicitely. Using these results, sufficient conditions are given on the means and the covariances relative to two equivalent Gaussian measures P and P sub lambda of a process X so that the Radon-Nikodym R-N derivative dP sub lambdadP is the exponential of the diagonal form in X. Analogues of the last two results in the set up of Hilbert space are also proved. Using one of these results, a simple proof of the integrability of the exponential of a certain multiple of the square of any continuous seminorm relative to Gaussian measure on separable nuclear space is given. Author
- Statistics and Probability