Properties of monotone functions of several variables and applications to limit theorems of probability theory

This paper provides some properties of monotone functions of several variables. It is well known that the set of discontinuity points of any monotone function of a single variable is at most countable. Generalization of this result for monotone functions of several variables is given. It is also established that it is possible to change the values of a monotone function of several variables at its discontinuity points to make the monotone function continuous from the right. These results are used to prove equivalence of convergence of a sequence of random vectors in distribution and pointwise convergence of the sequence of characteristic functions. A well-known consequence of this equivalence is the following result that is significant for econometrics. Convergence of a sequence of random vectors in distribution implies convergence of functions of the random vectors in distribution.