By Rene L Schilling; Renming Song; Zoran VondracМЊek
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Ii) g D e f where f is an extended Bernstein function. (iii) g is logarithmically completely monotone. Proof. (i))(ii) For c > 0 let gc . / WD g. C c/, 7! > 0. Then gc . / gc . c/ is again in CM, infinitely divisible and tends to 1 as ! 0. 7, there exists fc 2 BF such that g. C c/ D gc . c/e fc . / for all > 0. This can be written as g. / D g. c/e fc . c/ De fc . 5) If 0 < b < c the same formula is valid with b replacing c. In particular, for 0 < b < c < it holds that fc . c/ D fb . 0; 1/ ! R by f .
7! f1 . ˛ /f . ˇ / 2 Proof. 2). (ii) For every u > 0 we know that e ufn is a completely monotone function and that e uf . 1 e ufn . / . Since CM is closed under pointwise limits, cf. 6, e uf is completely monotone and f 2 BF. (iii) Let f1 ; f2 2 BF. g ı f1 / ı f2 2 CM. The converse direction (ii))(i) of Theorem shows that f1 ı f2 2 BF. 1 e /= D 0 e ds is completely monotone. Therefore, f. 6, completely monotone. 2). 2) and Fatou’s lemma. (vi) We know that the fractional powers 7! ˛ , 0 6 ˛ 6 1, are Bernstein ˛ ˇ functions.
N . If X denotes a random variable with distribution and Yj , 1 6 j 6 n, denote independent and identically distributed random variables with the common distribution n , this means that X and Y1 C C Yn have the same probability distribution. Depending on the structure of the triangular array, the limiting distributions may have different stability properties. We will discuss this for one-sided distributions supported in Œ0; 1/. Often it is easier to describe them in terms of the corresponding Laplace transforms.