Geodesic
On the accuracy of infinitely divisible approximation of $n$-fold convolutions of probability distributions
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 33, Tome 515 (2022), pp. 83-90

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Applying the results of Zaitsev (1987) to specific symmetric distributions with slowly decreasing power tails, we obtained power estimates for the accuracy of the infinitely divisible approximation of the distributions of sums of $n$ i.i.d. random variables of the form $O(n^{-1+\varepsilon})$ with $\varepsilon$ arbitrarily close to zero. 
@article{ZNSL_2022_515_a5,
     author = {Ya. S. Golikova and A. Yu. Zaitsev},
     title = {On the accuracy of infinitely divisible approximation of $n$-fold convolutions of probability distributions},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {83--90},
     publisher = {mathdoc},
     volume = {515},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2022_515_a5/}
}
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Ya. S. Golikova; A. Yu. Zaitsev. On the accuracy of infinitely divisible approximation of $n$-fold convolutions of probability distributions. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 33, Tome 515 (2022), pp. 83-90. http://geodesic.mathdoc.fr/item/ZNSL_2022_515_a5/