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)]=\sigm...

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Bibliographic Details
Published in:Theory of probability and its applications 2002, Vol.46 (4), p.699-707
Main Authors: CACOULLOS, T, PAPADATOS, N, PAPATHANASIOU, V
Format: Article
Language:English
Subjects:
Exact sciences and technology
Limit theorems
Mathematics
Probability and statistics
Probability theory and stochastic processes
Random variables
Sciences and techniques of general use
Stochastic processes
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Summary: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.
ISSN:0040-585X
1095-7219
DOI:10.1137/S0040585X97979366