arXiv subjects

Mathias Rafler

Publications and source records attributed to Mathias Rafler.

8 recordsLinked to original sources

arXiv2017

General thinning characterizations of distributions and point processes

For general thinning procedures, its inverse operation, the condensing, is studied and a link to integration-by-parts formulas is established. This extends the recent results on that link for independent thinnings of point processes to general thinnings of finite point processes. In particular, the classical integration-by-parts formulas appear as the example of independent thinnings. Moreover, the representation of the splitting kernel of finite point processes in terms of its reduced Palm kernels is extended to general thinnings. This link is studied in the context of discrete random variables and yields analogue characterizations of their distributions. Results on independent thinnings are complemented by a discrete stick breaking characterization of distributions.

math.PR↗

arXiv2014

The logical postulates of Böge, Carnap and Johnson in the context of Papangelou processes

We adapt Johnson's sufficiency postulate, Carnap's prediction invariance postulate and Böge's learn-merge invariance to the context of Papangelou processes and discuss equivalence of their generalizations, in particular their weak and strong generalizations. This discussion identifies a condition which occurs in the construction of Papangelou processes. In particular, we show that these generalizations characterize classes of Poisson and Pólya point processes.

math.PR↗

arXiv2013

Exit spaces for Cox processes and the Pólya sum process

For Cox processes we construct a Markov process with increasing paths to couple the condensations of the Cox process in a monotone way. A similar procedure procedure yields an analogue Markov process for the Pólya sum process. Moreover, we identify the exit spaces of these Markov processes and identify them firstly as mixtures of certain extremal processes, i.e. as a process in a random environment, and secondly as Gibbs processes.

math.PR↗

arXiv2013

A hydrodynamic limit and fluctuations for a Chinese restaurant-like process

We study a Markov process constructed from the Pólya sum process, which yields a kind of spatial version of the Chinese restaurant process, where each 'table' is assigned a 'location'. This construction firstly allows a definition of summation of independent processes, and secondly a monotone coupling for different parameters. Moreover, the process is related to a multi-particle random walk on the positive integers and its hydrodynamic limit and fluctuations are discussed.

math.PR↗

arXiv2013

The Polya branching process and limit theorems for conditioned random fields

The first aim is to construct generalizations of Polya type point process by applying a branching mechanism to these point processes. Conditions are given under which these point processes satisfy an integration by parts formula. Furthermore we compute their Palm kernels, which turn out to be superpositions of different point processes. Secondly we identify all point processes whose local characteristics agree with these of a fixed branching of a Polya type point process as mixtures of branchings of Polya type point process and show that in this case also they are characterized by an integration by parts formula.

math.PR↗

arXiv2012

Hoppe trees, random recursive sets and their barycentre

We consider a recursively defined random set of points and its barycenter, where the random set is constructed by the following inductive rule: Given a realization of $n-1$ points, one of them is picked at random and serves as a source the $n$-th point. We discuss the asymptotic behaviour of the barycentre of this random set. The main analysis relies on the analsis of Hoppe trees, for which we derive a limit theorem for the joint distribution of total length and Wiener index.

math.PR↗

arXiv2012

The Pólya sum kernel and Bayes estimation

We consider a particular Cox process from a Bayesian viewpoint and show that the Bayes estimator of the intensity measure is the so-called Pólya sum kernel, which occurred recently in the context of the construction of the so-called Papangelou processes. More precisely, if the prior, the directing measure of the Cox process, is a Poisson-Gamma random measure, then the posterior is again a Poisson-Gamma random measure and the Bayes estimator of the intensity is the Pólya sum kernel. Moreover, we extend this result to doubly stochastic Poisson-Gamma priors and give conditions under which one can identify the Bayes estimator for the intensity.

math.PR↗

arXiv2012

The Pólya sum process: Limit theorems for conditioned random fields

In \cite{hZ09}, Zessin constructed the so-called Pólya sum process via partial integration technique. This process shares some important properties with the Poisson process such as complete randomness and infinite divisibility. This work discusses H-sufficient statistics for the Pólya sum process as it was done for the Poisson process in \cite{hZ76}.

math.PR↗