Geodesic
On the non-uniform estimate of the rate of convergence in the local limit theorem in the case of a~stable limit distribution
Teoriâ veroâtnostej i ee primeneniâ, Tome 27 (1982) no. 3, pp. 566-568

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Let $p_n(x)$ be a probability density function of normalized and centered sum of $n$ i. i. d. random variables belonging to the domain of attraction of the stable distribution $G$ of index $\alpha$, $0\alpha\le 2$, $\alpha\ne 1$. Let $p(x)$ be a probability density function of $G$. It is proved that under certain conditions the relation $$ \lim_{n\to\infty}|x|^\delta|p_n(x)-p(x)|=0,\qquad 0\le\delta\alpha\ne 1, $$ holds uniformly in $x$, $-\infty$. 
@article{TVP_1982_27_3_a13,
     author = {A. S. Mal'kov and V. V. Ul'yanov},
     title = {On the non-uniform estimate of the rate of convergence in the local limit theorem in the case of a~stable limit distribution},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {566--568},
     publisher = {mathdoc},
     volume = {27},
     number = {3},
     year = {1982},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1982_27_3_a13/}
}
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A. S. Mal'kov; V. V. Ul'yanov. On the non-uniform estimate of the rate of convergence in the local limit theorem in the case of a~stable limit distribution. Teoriâ veroâtnostej i ee primeneniâ, Tome 27 (1982) no. 3, pp. 566-568. http://geodesic.mathdoc.fr/item/TVP_1982_27_3_a13/