A Natural L(,p)-Metric for Spaces Composed of Probability Measures With P-Th Moment.

Abstract

In the dissertation, it is seen that every probability measure with p-th moment on a complete, separable metric space can be viewed as a distribution of a metric space valued random variable. Between such random variables there exists an L(,p)-distance, and by finding the infimum of the L(,p)-distances between two types of random variables, it is possible to define a distance between distributions. It is seen that this distance can serve as a complete metric on the space of probability measures with p-th moment. The topology produced is shown to be equivalent to the topology of weak convergence of measures with convergence of moments. A closed embedding of this space into the space of finite measures with the weak convergence topology is produced and used to transfer Prokhorov's theorem on tightness and relative compactness to the new setting. Other results, including a computational method for the case 1 (LESSTHEQ) p (LESSTHEQ) (INFIN) on the set of real numbers are also detailed and proven.