The Law of Large Numbers in a Metric Space with a Convex Combination Operation

We consider a separable complete metric space equipped with a convex combination operation. For such spaces, we identify the corresponding convexification operator and show that the invariant elements for this operator appear naturally as limits in the strong law of large numbers. It is shown how to uplift the suggested construction to work with subsets of the basic space in order to develop a systematic way of proving laws of large numbers for such operations with random set

The structure of classical extensions of quantum probability theory

On the basis of a suggestive definition of a classical extension of quantum mechanics in terms of statistical models, we prove that every such classical extension is essentially given by the so-called Misra–Bugajski reduction map. We consider how this map enables one to understand quantum mechanics as a reduced classical statistical theory on the projective Hilbert space as phase space and discuss features of the induced hidden-variable model. Moreover, some relevant technical results on the topology and Borel structure of the projective Hilbert space are reviewed

Representation of maxitive measures: an overview

Idempotent integration is an analogue of Lebesgue integration where $\sigma$-maxitive measures replace $\sigma$-additive measures. In addition to reviewing and unifying several Radon--Nikodym like theorems proven in the literature for the idempotent integral, we also prove new results of the same kind.Comment: 40 page

Convergence and Rates for Fixed-Interval Multiple-Track Smoothing Using kk-Means Type Optimization

We address the task of estimating multiple trajectories from unlabeled data. This problem arises in many settings, one could think of the construction of maps of transport networks from passive observation of travellers, or the reconstruction of the behaviour of uncooperative vehicles from external observations, for example. There are two coupled problems. The first is a data association problem: how to map data points onto individual trajectories. The second is, given a solution to the data association problem, to estimate those trajectories. We construct estimators as a solution to a regularized variational problem (to which approximate solutions can be obtained via the simple, efficient and widespread $k$-means method) and show that, as the number of data points, $n$, increases, these estimators exhibit stable behaviour. More precisely, we show that they converge in an appropriate Sobolev space in probability and with rate $n^{-1/2}$

Stability and statistical inferences in the space of topological spatial relationships

Modelling topological properties of the spatial relationship between objects, known as the extit{topological relationship}, represents a fundamental research problem in many domains including Artificial Intelligence (AI) and Geographical Information Science (GIS). Real world data is generally finite and exhibits uncertainty. Therefore, when attempting to model topological relationships from such data it is useful to do so in a manner which is both extit{stable} and facilitates extit{statistical inferences}. Current models of the topological relationships do not exhibit either of these properties. We propose a novel model of topological relationships between objects in the Euclidean plane which encodes topological information regarding connected components and holes. Specifically, a representation of the persistent homology, known as a persistence scale space, is used. This representation forms a Banach space that is stable and, as a consequence of the fact that it obeys the strong law of large numbers and the central limit theorem, facilitates statistical inferences. The utility of this model is demonstrated through a number of experiments