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
