An Application of a Density Transform and the Local Limit Theorem

Abstract

Consider an absolutely continuous random variable X with finite variance~$\sigma^2$. 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 ${\rm Cov}[X,g(X)]=\sigma^2 \mathbf{E} [g'(X^*)]$ for any absolutely continuous function g with derivative $g'$, provided that $\mathbf{E} |g'(X^*)| < \infty$. 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.