Abstract
Let ξ be a random varable which can be written in the form
where each of ξ1, ξ2, ..., ξ N is a sum of independent random variables, and, p 1, p 2, ..., p N are nonrandom numbers such that p 1 + p 2 + p N = 1, p i > 0. Note that, generally speaking, ξ1, ξ2, ..., ξ N are not independent random variables. For any s > 0, according to Chebyshev’s inequality, we get
under the condition that Eexp(sξ) < ∞, 0 < s < s 0. By Jensen’s inequality, we obtain exp(sξ) ≤ p 1 exp(sξ1) + ... + p N exp(sξ N ). Therefore, we have
for t > 0.
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© 1994 Springer Science+Business Media Dordrecht
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Koroljuk, V.S., Borovskich, Y.V. (1994). Probabilities of Large Deviations. In: Theory of U-Statistics. Mathematics and Its Applications, vol 273. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-3515-5_9
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DOI: https://doi.org/10.1007/978-94-017-3515-5_9
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-4346-7
Online ISBN: 978-94-017-3515-5
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