By Ulf Grenander
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Additional resources for Abstract Inference [math. stats.]
Of a subordinator S are non-decreasing functions. e. ea > t/ D e at , t > 0. We allow that a D 0 in which case ea D C1. Assume that ea is independent of the subordinator S . 2) C1; t > ea : The process b S is the subordinator S killed at an independent exponential time. Any process with state space Œ0; 1 having the same distribution as b S will be called a killed subordinator. The connection between (killed) subordinators and convolution semigroups of subprobability measures on Œ0; 1/ is as follows.
Ds/ dt 0 D 1 ; f. 12) 5 A probabilistic intermezzo 45 where f 2 BF is the Bernstein function corresponding to the convolution semigroup . t / t >0 . dt/ > e U Œ0; ; Œ0;1/ Œ0; proving that U is finite on bounded sets. 16. 0; 1/ ! 0; 1/ is said to be a potential if f D 1=g where g 2 BF D BF n ¹0º. The set of all potentials will be denoted by P. 12), P consists of Laplace transforms of potential measures. e. 17. Let g 2 P be a potential. Then g is logarithmically completely monotone and hence there exists an extended Bernstein function f such that g D e f .
0C/ 6 1, the sub-probability measure on Œ0; 1/ satisfying L D g is said to be an infinitely divisible distribution; we write 2 ID. The discussion preceding the definition shows that if f 2 BF, then g WD e f is completely monotone and infinitely divisible. 0C/ 6 1. This is already one direction of the next result. 7. 0; 1/ ! 0; 1/. Then the following statements are equivalent. 0C/ 6 1. (ii) g D e f where f 2 BF. 38 5 A probabilistic intermezzo Proof. Suppose that (i) holds. 0C/ 6 1, there exists a sub-probability measure t on Œ0; 1/ such that g t .