 # On subordinators, self-similar Markov processes and some factorizations of the exponential variable

Bertoin, Jean; Yor, Marc (2001). On subordinators, self-similar Markov processes and some factorizations of the exponential variable. Electronic Communications in Probability, 6:95-106.

## Abstract

Let ξ be a subordinator with Laplace exponent Φ, I=∫∞0exp(−ξs)ds the so-called exponential functional, and X (respectively, X^) the self-similar Markov process obtained from ξ (respectively, from ξ^=−ξ) by Lamperti's transformation. We establish the existence of a unique probability measure ρ on ]0,∞[ with k-th moment given for every k∈N by the product Φ(1)⋯Φ(k), and which bears some remarkable connections with the preceding variables. In particular we show that if R is an independent random variable with law ρ then IR is a standard exponential variable, that the function t→E(1/Xt) coincides with the Laplace transform of ρ, and that ρ is the 1-invariant distribution of the sub-markovian process X^. A number of known factorizations of an exponential variable are shown to be of the preceding form IR for various subordinators ξ.

## Abstract

Let ξ be a subordinator with Laplace exponent Φ, I=∫∞0exp(−ξs)ds the so-called exponential functional, and X (respectively, X^) the self-similar Markov process obtained from ξ (respectively, from ξ^=−ξ) by Lamperti's transformation. We establish the existence of a unique probability measure ρ on ]0,∞[ with k-th moment given for every k∈N by the product Φ(1)⋯Φ(k), and which bears some remarkable connections with the preceding variables. In particular we show that if R is an independent random variable with law ρ then IR is a standard exponential variable, that the function t→E(1/Xt) coincides with the Laplace transform of ρ, and that ρ is the 1-invariant distribution of the sub-markovian process X^. A number of known factorizations of an exponential variable are shown to be of the preceding form IR for various subordinators ξ.

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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics Physical Sciences > Statistics and Probability Social Sciences & Humanities > Statistics, Probability and Uncertainty English 2001 24 Jul 2013 08:14 30 Jul 2020 09:26 Institute of Mathematical Statistics 1083-589X Hybrid Publisher DOI. An embargo period may apply. https://doi.org/10.1214/ECP.v6-1039 http://www.ams.org/mathscinet-getitem?mr=1871698http://www.zentralblatt-math.org/zbmath/search/?q=an%3A1024.60030

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