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Transactions of the American Mathematical Society

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The semigroup of metric measure spaces and its infinitely divisible probability measures

Authors: Steven N. Evans and Ilya Molchanov
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 43A05, 60B15, 60E07, 60G51
Published electronically: May 3, 2016
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Abstract: A metric measure space is a complete, separable metric space equipped with a probability measure that has full support. Two such spaces are equivalent if they are isometric as metric spaces via an isometry that maps the probability measure on the first space to the probability measure on the second. The resulting set of equivalence classes can be metrized with the Gromov-Prohorov metric of Greven, Pfaffelhuber and Winter. We consider the natural binary operation $ \boxplus $ on this space that takes two metric measure spaces and forms their Cartesian product equipped with the sum of the two metrics and the product of the two probability measures. We show that the metric measure spaces equipped with this operation form a cancellative, commutative, Polish semigroup with a translation invariant metric. There is an explicit family of continuous semicharacters that is extremely useful for, inter alia, establishing that there are no infinitely divisible elements and that each element has a unique factorization into prime elements.

We investigate the interaction between the semigroup structure and the natural action of the positive real numbers on this space that arises from scaling the metric. For example, we show that for any given positive real numbers $ a,b,c$ the trivial space is the only space $ \mathcal {X}$ that satisfies $ a \mathcal {X} \boxplus b \mathcal {X} = c \mathcal {X}$.

We establish that there is no analogue of the law of large numbers: if $ \mathbf {X}_1, \mathbf {X}_2, \ldots $ is an identically distributed independent sequence of random spaces, then no subsequence of $ \frac {1}{n} \bigboxplus _{k=1}^n \mathbf {X}_k$ converges in distribution unless each $ \mathbf {X}_k$ is almost surely equal to the trivial space. We characterize the infinitely divisible probability measures and the Lévy processes on this semigroup, characterize the stable probability measures and establish a counterpart of the LePage representation for the latter class.

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Additional Information

Steven N. Evans
Affiliation: Department of Statistics #3860, 367 Evans Hall, University of California, Berkeley, California 94720-3860

Ilya Molchanov
Affiliation: Institute of Mathematical Statistics and Actuarial Science, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland

Keywords: Gromov--Prohorov metric, cancellative semigroup, monoid, Delphic semigroup, semicharacter, irreducible, prime, unique factorization, L\'evy--Hincin formula, It\^o representation, L\'evy process, stable probability measure, LePage representation, law of large numbers
Received by editor(s): January 27, 2014
Received by editor(s) in revised form: September 17, 2014, and March 9, 2015
Published electronically: May 3, 2016
Additional Notes: The first author was supported in part by NSF grant DMS-09-07630 and NIH grant 1R01GM109454-01
The second author was supported in part by Swiss National Science Foundation grant 200021-137527
Article copyright: © Copyright 2016 American Mathematical Society