Abstract
We provide precise bounds for tail probabilities, say ℙ{M n ⩾ x}, of sums M n of bounded i.i.d. random variables. The bounds are expressed through tail probabilities of sums of i.i.d. Bernoulli random variables. In other words, we show that the tails are sub-Bernoullian. Sub-Bernoullian tails are dominated by Gaussian tails. Possible extensions of the methods are discussed.
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References
G. R. Shorack and J. A. Wellner, Empirical Processes with Applications to Statistics, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, Wiley, New York (1986).
V. V. Petrov, Sums of Independent Random Variables, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 82, Springer, New York(1975).
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Bentkus, V. An Inequality for Large Deviation Probabilities of Sums of Bounded i.i.d. Random Variables. Lithuanian Mathematical Journal 41, 112–119 (2001). https://doi.org/10.1023/A:1011668015105
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DOI: https://doi.org/10.1023/A:1011668015105