I believe the following book by Yakov Sinai: "Probability Theory  An Introductory Course" (Springer)  uses such an argument to proove a version of the central limit theorem in chaper 15.
UPDATE
A better reference is the updated verson of that book, Leonid Koralov, Yakov Sinai:  "Theory of Probability and Random Prosecces, second edition" (Universitext).