Geodesic
On uniform convergence of the frequencies of events to their probabilities
Teoriâ veroâtnostej i ee primeneniâ, Tome 16 (1971) no. 2, pp. 264-279

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The paper considers the problem of uniform (over a class of events) convergence in probability of the frequences of events to their probabilities. Some sufficient conditions of the convergence are found which are expressed in terms of the class of events considered only. Necessary and sufficient conditions are also obtained which take into account thebasic probability measure properties. 
@article{TVP_1971_16_2_a4,
     author = {V. N. Vapnik and A. Ya. \v{C}ervonenkis},
     title = {On uniform convergence of the frequencies of events to their probabilities},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {264--279},
     publisher = {mathdoc},
     volume = {16},
     number = {2},
     year = {1971},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1971_16_2_a4/}
}
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V. N. Vapnik; A. Ya. Červonenkis. On uniform convergence of the frequencies of events to their probabilities. Teoriâ veroâtnostej i ee primeneniâ, Tome 16 (1971) no. 2, pp. 264-279. http://geodesic.mathdoc.fr/item/TVP_1971_16_2_a4/