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

January, 2014
Report Number: 
Steven N. Evans and Ilya Molchanov
A compact metric measure space is a compact metric space equipped   with 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 compact 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 compact metric measure spaces equipped with this   operation form a cancellative, commutative, Polish semigroup   with a translation invariant metric   and that each element has a unique factorization into prime   elements.  Moreover, there is an explicit family of continuous   semicharacters that are extremely useful in understanding the   properties of this semigroup.     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 L\'evy 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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