Geodesic
Small deviation probabilities of weighted sum of independent random variables with a common distribution having a power decrease in zero under minimal moment assumptions
Teoriâ veroâtnostej i ee primeneniâ, Tome 62 (2017) no. 3, pp. 610-616

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We represent a certain weight condition such that the exact formula for the asymptotic behavior of small deviation probabilities for weighted sum of independent random variables with a common distribution, which has a power decay at zero, is valid under optimal moment assumptions.
Keywords: small deviations, sums of independent positive random variables, slowly varying functions. 
@article{TVP_2017_62_3_a8,
     author = {L. V. Rozovskii},
     title = {Small deviation probabilities of weighted sum of independent random variables with a common distribution having a power decrease in zero under minimal moment assumptions},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {610--616},
     publisher = {mathdoc},
     volume = {62},
     number = {3},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2017_62_3_a8/}
}
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L. V. Rozovskii. Small deviation probabilities of weighted sum of independent random variables with a common distribution having a power decrease in zero under minimal moment assumptions. Teoriâ veroâtnostej i ee primeneniâ, Tome 62 (2017) no. 3, pp. 610-616. http://geodesic.mathdoc.fr/item/TVP_2017_62_3_a8/