By Rene L Schilling; Renming Song; Zoran VondracМЊek
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Are completely monotone. 6(ii) the functions f10 . ˛ /; f20 . ˇ /; f1 . ˛ /= ˛ and f2 . ˇ /= ˇ 3 Bernstein functions 21 are again completely monotone. Since ˛ C ˇ 6 1, 7! ˛Cˇ 1 is completely monotone. 6, h0 is completely monotone. 7 says that on the set BF the notions of pointwise convergence, locally uniform convergence, and even convergence in the space C 1 coincide. 8. 1 fn . / D f . 0; 1/. 1 fn . k/ . 0; 1/. 0; / In both formulae we may replace lim infn by lim supn . Proof. 7(ii) we know that f 2 BF.
2. 19. We check first that each Á t is a sub-probability measure. du/ Œ0;1/ L . sI / Z D u e sg. ds/ Œ0;1/ DL t I g. g. g. g. g. Ás I /; as well as vague continuity since t 7! g. // is continuous. 0;1/ e ˛s L . 0;1/ Z D e Ã t sg. 1 sg. 1 sg. 1 e t /, t > 0, with a suitable constant c ;u > 0. 0; 1/, cf. 3. Now we have Z f g. / D a C bg. 1 e tg. 21. A good treatment of convolution semigroups and their potential theory is the monograph by Berg and Forst ; the classic exposition by Feller, , covers the material from a probabilistic angle and is still readable.
The name self-decomposable appears for the first time in 1955 in the first edition of Loève’s book on probability theory , the notation L goes back to Khintchine , see the comment in Gnedenko and Kolmogorov [108, Section 29]. The best sources for potential theory of convolution semigroups are Berg and Forst [29, Chapter III] and the original papers by F. Hirsch from the early 1970s. A more probabilistic approach can be found in Bertoin [36, Chapter II, III]. A somewhat hard to read but otherwise most comprehensive treatment is the monograph by Dellacherie and Meyer [77, 78].
Bernstein functions : theory and applications by Rene L Schilling; Renming Song; Zoran VondracМЊek