Geodesic
A generalization of the Chung-Erd\"os inequality for the probability of the union of events
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 11, Tome 341 (2007), pp. 147-150

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A generalization of the Chung–Erdös inequality for the probability of the union of arbitrary events is proved using some lower bounds for tail probabilities. We present a lower bound for the probability of appearance of at least $m$ events from the set of events $A_1,\dots,A_n$ where $1\le m\le n$. 
@article{ZNSL_2007_341_a9,
     author = {V. V. Petrov},
     title = {A generalization of the {Chung-Erd\"os} inequality for the probability of the union of events},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {147--150},
     publisher = {mathdoc},
     volume = {341},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2007_341_a9/}
}
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V. V. Petrov. A generalization of the Chung-Erd\"os inequality for the probability of the union of events. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 11, Tome 341 (2007), pp. 147-150. http://geodesic.mathdoc.fr/item/ZNSL_2007_341_a9/