Random Probability Measures on Polish Spaces

This monograph provides a thorough, comprehensive investigation of the narrow topology on random probability measures on Polish spaces. As a special feature, the author makes no additional assumptions on the probability space in the background, such as completeness or a countable generated algebra. One of the main results presented is a direct proof of the random analog of the Prohorov theorem, which is obtained without invoking an embedding of the Polish space into a compact space. Random Probability Measures on Polish Spaces is a clear and incisive volume that will prove useful for graduate students and researchers in mathematical analysis and its applications.

In this monograph the narrow topology on random probability measures on Polish spaces is investigated in a thorough and comprehensive way. As a special feature, no additional assumptions on the probability space in the background, such as completeness or a countable generated algebra, are made. One of the main results is a direct proof of the random analog of the Prohorov theorem, which is obtained without invoking an embedding of the Polish space into a compact space. Further, the narrow topology is examined and other natural topologies on random measures are compared. In addition, it is shown that the topology of convergence in law-which relates to the "statistical equilibrium"-and the narrow topology are incompatible. A brief section on random sets on Polish spaces provides the fundamentals of this theory. In a final section, the results are applied to random dynamical systems to obtain existence results for invariant measures on compact random sets, as well as uniformity results in the individual ergodic theorem. This clear and incisive volume is useful for graduate students and researchers in mathematical analysis and its applications.