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6 and comments after its proof in this reference. d. with common distribution F , let Fn be the empirical distribution of X1 , . . , Xn , n ∈ N, and let Fn−1 denote the empirical quantile function. d. uniform on [0, 1]. 8) n (t) , where G−1 n is the quantile function corresponding to the uniform variables U1 , . . , Un . 9) vn (t) := n Fn−1 (t) − F −1 (t) , n ∈ N. 10) for some ξ between t and G−1 n (t). This Taylor expansion can be used to relate the (weighted) L2 norms of vn and un /f (F −1 ).

S. We should point out that the results that follow do not require the variables ξi to be exponential, but only to be integrable enough, however, we stay with exponential variables, which is what we need. 4 ) of Ln are allowed to be arbitrary functions in L2 (0, 1). Given a triangular array cn,i , i ≤ n, n ∈ N, of functions in L2 (0, 1), we set n Yn (t) := cn,i ξi , Zn (t) = i=1 n−1 Sn 2 Yn (t), t ∈ [0, 1]. 19) Define cn,i,j = ci,j as 1 cn,i (t)cn,j (t)dt := cn,i , cn,j , i, j = 1, . . , n, n ∈ N.

On the opposite side, some examples of symmetric distributions, with shapes very far from normality (some of them even multimodal),√and β2 = 3 are known (see, for instance, [4, 52]). As a consequence, none of the b1 , b2 , K 2 or R tests detects the non-normality of the parent distribution in all cases. Empirical and Quantile Processes 31 Other tests of normality are the u-test [25], based on the ratio between the range and the standard deviation in the sample, and the a-test [45], which studies the ratio of the sample mean to the standard deviation.

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