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updating the answer

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).