An application of a density transform and the local limit theorem

Abstract

Consider an absolutely continuous random variable X with finite variance σ2. It is known that there exists another random variable X* (which can be viewed as a transformation of X) with a unimodal density, satisfying the extended Stein-type covariance identity Cov[X, g(X)] = σ2E[g′(X*)] for any absolutely continuous function g with derivative g′, provided that E|g′(X*)| < ∞. Using this transformation, upper bounds for the total variation distance between two absolutely continuous random variables X and Y are obtained. Finally, as an application, a proof of the local limit theorem for sums of independent identically distributed random variables is derived in its full generality.