The relation between the closeness of random variables and their distributions

Back to the series of Problems in Probability Theory

In this series of problem the following problem is investigated. If two probability measures are given on a separable metric space, then we want to construct such random variables with these distributions which are as close to each other as it is possible. We show that this question is in close relation to a classical problem of probability theory. The Prochorov distance of probability measures is investigated. It is shown that if a sequence of probability measures weakly converges, then there exists such sequence of random variables with such distributions which is convergent with probability one. The quantile transform is investigated, and its most important properties are proved.

At the end the invariance principle (functional central limit theorem) is proved by means of coupling arguments.

33 pages