Geodesic
Small deviation probabilities for a weighted sum of independent positive random variables with common distribution function that can decrease at zero fast enough
Teoriâ veroâtnostej i ee primeneniâ, Tome 63 (2018) no. 1, pp. 191-202

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The present note puts forward the most general conditions to date under which the classical asymptotics of the small deviation probabilities for a weighted sum of independent positive random variables takes place. Regarding random variables it is assumed that their common distribution function can, in particular, have an exponential falloff at zero.
Keywords: small deviations, sums of independent positive random variables. 
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     author = {L. V. Rozovskii},
     title = {Small deviation probabilities for a weighted sum of independent positive random variables with common distribution function that can decrease at zero fast enough},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {191--202},
     publisher = {mathdoc},
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     number = {1},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2018_63_1_a9/}
}
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L. V. Rozovskii. Small deviation probabilities for a weighted sum of independent positive random variables with common distribution function that can decrease at zero fast enough. Teoriâ veroâtnostej i ee primeneniâ, Tome 63 (2018) no. 1, pp. 191-202. http://geodesic.mathdoc.fr/item/TVP_2018_63_1_a9/