Geodesic
Small deviation probabilities for sums of independent positive random variables
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 11, Tome 341 (2007), pp. 151-167

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In the note we give estimates of small deviation probabilities of a sum $\sum_{j\ge 1} \lambda_j X_j$, where $\{\lambda_j\}$ are nonnegative numbers and $\{X_j\}$ are i.i.d. positive random variables, satisfying mild assumptions at zero and infinity. 
@article{ZNSL_2007_341_a10,
     author = {L. V. Rozovskii},
     title = {Small deviation probabilities for sums of independent positive random variables},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {151--167},
     publisher = {mathdoc},
     volume = {341},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2007_341_a10/}
}
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L. V. Rozovskii. Small deviation probabilities for sums of independent positive random variables. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 11, Tome 341 (2007), pp. 151-167. http://geodesic.mathdoc.fr/item/ZNSL_2007_341_a10/