An Application of a Density Transform and the Local Limit Theorem

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.