Geodesic
Limit distributions of a statistical estimate for the entropy
Teoriâ veroâtnostej i ee primeneniâ, Tome 18 (1973) no. 3, pp. 643-650 
A. M. Zubkov. Limit distributions of a statistical estimate for the entropy. Teoriâ veroâtnostej i ee primeneniâ, Tome 18 (1973) no. 3, pp. 643-650. http://geodesic.mathdoc.fr/item/TVP_1973_18_3_a23/
@article{TVP_1973_18_3_a23,
     author = {A. M. Zubkov},
     title = {Limit distributions of a~statistical estimate for the entropy},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {643--650},
     year = {1973},
     volume = {18},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1973_18_3_a23/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let $\widehat p_j$ be the relative frequency with which the event $E_j$ occurs in $N$ mutually independent identically distributed trials ($j=1,2,\dots,s$; $E_i\cap E_j=\varnothing$, $i\ne j$; $\mathbf P\{\bigcup\limits_{j=1}^sE_j\}=1$). We find conditions for the estimate $$ H=-\sum_{j=1}^s\widehat p_j\ln\widehat p_j $$ of the entropy $H=-\sum_{j=1}^s\mathbf P\{E_j\}\ln\mathbf P\{E_j\}$ to be asymptotically normal (as $N,s\to\infty$) or asymptotically $\chi^2$ (as $N\to\infty$, $s$ fixed).