On Khintchine's Theorem and its Place in Random Variate Generation

Abstract

It is well known that if X and Y are uniformly distributed over the region between the horizontal axis and a density function f, then X is distributed according to density f. The algorithm “generate Y from its marginal distribution, then X from its uniform conditional distribution given Y == y” follows. The main point made in this article is that for monotone and unimodal distributions, this construction reduces to Khintchine's theorem, thereby yielding a simple explication thereof. This observation is followed up with further consideration of the general, nonunimodal, case for both univariate and multivariate distributions, and parallels are drawn with an alternative random variate generation method called vertical density representation.