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

Report Number

823

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.

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