Exercises in probability

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Random Fields and Geometry (2007, background): “We shall, however, take a more direct route, which we learnt about from the excellent collection of exercises [37], although its roots are much older.”
Lamperti-type laws (2007, methodology): “ 4 π |x| , −∞<x<∞. Remark 2.5. Note that the characteristic function of απ−1 log(Xα) is equivalent to E[eiαλ/πlog(Xα)]= sinh(αλ) αsinh(λ) . This expression can be found in Chaumont and Yor ([20], page 147). In addition we see that this characteristic function agrees with the class of generali”
Limit distributions for large P\'{o}lya urns (2009, background): “A probability distribution A is called invertible when, for any probability distributions A and B, the equation AX = B admits a unique solution X independent of A, see, for instance, Chaumont and Yor [4]. The invertibility of any power of a Gamma distribution can be shown by elementary consideration”