Geodesic
An Application of a Density Transform and the Local Limit Theorem
Teoriâ veroâtnostej i ee primeneniâ, Tome 46 (2001) no. 4, pp. 803-810

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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.
Keywords: density transform, local limit theorem for densities.
Mots-clés : total variation distance 
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T. Cacoullos; N. Papadatos; V. Papathanasiou. An Application of a Density Transform and the Local Limit Theorem. Teoriâ veroâtnostej i ee primeneniâ, Tome 46 (2001) no. 4, pp. 803-810. http://geodesic.mathdoc.fr/item/TVP_2001_46_4_a13/