Random sequences in Fr�chet spaces

Abstract

This article deals with the generation of arbitrarily distributed sequencesΦ of random variables in a Frechet space, using sequences ofcanonical random variables (c.r.v.)-i.e., independently uniformly distributed random variables taking real values in the unit interval [0, 1)-orcanonical random digits (c.r.d.)-i.e., independently uniformly distributed random variables taking integer values in some finite interval [0,B−1]. Two main results are established. First, that the members of a sequence of real random variables in [0, 1) are c.r.v. if and only if all the digits of all thebase- B digital representations of the members of the sequence are c.r.d. Secondly, that, given any sequenceΦ of random variables in a Frechet space, there is a sequenceΨ of functionsψn(ξ1,ξ2, ...,ξn), forn=1, 2, 3,... (whereξ1,ξ2,...,ξn,... are c.r.v.) which is distributed identically toΦ.