Geodesic
A local limit theorem for unequally distributed random variables
Teoriâ veroâtnostej i ee primeneniâ, Tome 13 (1968) no. 2, pp. 348-351 
V. M. Kruglov. A local limit theorem for unequally distributed random variables. Teoriâ veroâtnostej i ee primeneniâ, Tome 13 (1968) no. 2, pp. 348-351. http://geodesic.mathdoc.fr/item/TVP_1968_13_2_a14/
@article{TVP_1968_13_2_a14,
     author = {V. M. Kruglov},
     title = {A~local limit theorem for unequally distributed random variables},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {348--351},
     year = {1968},
     volume = {13},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1968_13_2_a14/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let $\xi_1,\dots,\xi_n$ be a sequence of independent random variables. Form another sequence $$ \eta_n=\frac{\xi_1+\dots+\xi_n}{B_n}-A_n.\eqno(1) $$ Suppose that for any $n$ $\xi_n$ has one of $\tau$ absolutely continuous distributions $$ F_1(x),F_2(x),\dots,F_\tau(x) $$ The following assertion is proved. For the sequence of the densities $p_n(x)$ of the sums (1) to converge uniformly to the density of a limit law for some $B_n>0$, $A_n$ it is necessary and sufficient that 1. $\mathbf P\{\eta_n weakly ($G$ is the limit law). 2. There exists such an $N$ that $p_N(x)$ is bounded.