By F. Hirsch, G. Mokobodzki, M. Brelot, G. Choquet, J. Deny

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**Sample text**

Moreover, if Z is distributed as a normal N (0, 1) random variable, then Z 2 has the Γ (1/2, 2) distribution. Consequently, sums of Gamma random variables include sums of exponential random variables as well as sums of weighted chi-square random variables. As shown by the lemma below, the Gamma random variables have a nice Laplace transform. 55. Let X be a random variable with Γ (a, b) distribution. 101) where is the strictly convex function given by (t) = − log(1 − t) if t < 1, +∞ if t 1. 56. 55 that E[X] = ab (0) = ab and Var(X) = ab2 (0) = ab2 .

Hence, 2Vn + Wn (5Vn + Vn )/3 2Vn . 82). The inequality with the denominator 2Vn may be found in Maurer [16]. 43. Let X1 , . . , Xn be a finite sequence of independent random variables sharing the same Exponential E (λ ) distribution with λ > 0. In that case, Wn = 0, which ensures that for any positive x, P(Sn E[Sn ] − x) exp − x2 . 7, we will give more efficient inequalities for sums of independent random variables with exponential distributions. This is the reason why we give a second example below.

55 that, for any real t, log E[exp(t(Sn − E[Sn ]))] n ∑ ak c (bk t). 109) k=1 Moreover, let hc be the function defined, for any positive t, by hc (t) = c (t)/t 2 . The function hc is increasing on ]0, +∞[. Hence, for any positive t, c (bk t) b2k c (t), which implies that log E[exp(t(Sn − E[Sn ]))] b 2 2,a c (t). 110) We deduce from Markov’s inequality that for any positive x and for any t in ]0, 1[, log P(Sn − E[Sn ] x b 2 2,a ) − b 2 2,a (xt − c (t)). 111) The optimal value t in the above inequality is given by the elementary equation c (t) = t/(1 − t) = x, leading to t = x/(1 + x).