Geodesic
Remarks on the Markov--Krein identity and quasi-invariance of the gamma process
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part VI, Tome 283 (2001), pp. 21-36

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We present a simple proof of the Markov–Krein identity for distributions of means of linear functionals of the Dirichlet process and its various generalizations. The key idea is to use the representation of the Dirichlet process as the normalized gamma process and fundamental properties of gamma processes. 
@article{ZNSL_2001_283_a2,
     author = {A. M. Vershik and M. Yor and N. V. Tsilevich},
     title = {Remarks on the {Markov--Krein} identity and quasi-invariance of the gamma process},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {21--36},
     publisher = {mathdoc},
     volume = {283},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2001_283_a2/}
}
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A. M. Vershik; M. Yor; N. V. Tsilevich. Remarks on the Markov--Krein identity and quasi-invariance of the gamma process. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part VI, Tome 283 (2001), pp. 21-36. http://geodesic.mathdoc.fr/item/ZNSL_2001_283_a2/