Geodesic
The analogue of the law of large numbers for additive functions on sparse sets
Matematičeskie zametki, Tome 18 (1975) no. 5, pp. 687-698 Cet article a éte moissonné depuis la source Math-Net.Ru

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An analog of the Turan'n–Kubilyus inequality is proved for a sufficiently wide class of sequences which contains, in particular, $a_n=f(n)$ and $a_n=f(p_n)$, where $f(n)$ is a polynomial with integral coefficients. This result helps us to obtain integral limit theorems for additive functions on the class of sequences under investigation. 
@article{MZM_1975_18_5_a4,
     author = {B. V. Levin and N. M. Timofeev},
     title = {The analogue of the law of large numbers for additive functions on sparse sets},
     journal = {Matemati\v{c}eskie zametki},
     pages = {687--698},
     year = {1975},
     volume = {18},
     number = {5},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1975_18_5_a4/}
}
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B. V. Levin; N. M. Timofeev. The analogue of the law of large numbers for additive functions on sparse sets. Matematičeskie zametki, Tome 18 (1975) no. 5, pp. 687-698. http://geodesic.mathdoc.fr/item/MZM_1975_18_5_a4/