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**Term(s):**2014**Results:**1**Sorted by:**

**Title:**The semigroup of compact metric measure spaces and its infinitely divisible probability measures**Author(s):**Evans, Steven N.; Molchanov, Ilya; **Date issued:**January 2014

http://nma.berkeley.edu/ark:/28722/bk00153296s (PDF) **Abstract:**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.**Keyword note:**Evans__Steven_N Molchanov__Ilya**Report ID:**823**Relevance:**100