Anja Sturm

Voter models with heterozygosity selection.

This paper studies variations of the usual voter model that favor types that are locally less common. Such models are dual to certain systems of branching annihilating random walks that are parity preserving. For both the voter models and their dual branching annihilating systems we determine all homogeneous invariant laws, and we study convergence to these laws started from other initial laws.

Limit laws of the empirical Wasserstein distance

We derive central limit theorems for the Wasserstein distance between the empirical distributions of Gaussian samples. The cases are distinguished whether the underlying laws are the same or different. Results are based on the (quadratic) Frechet differentiability of the Wasserstein distance in the gaussian case. Extensions to elliptically symmetric distributions are discussed as well as several applications such as bootstrap and statistical testing.

On spatial coalescents with multiple mergers in two dimensions

We consider the genealogy of a sample of individuals taken from a spatially structured population when the variance of the offspring distribution is relatively large. The space is structured into discrete sites of a graph G. If the population size at each site is large, spatial coalescents with multiple mergers, so called spatial Λ-coalescents, for which ancestral lines migrate in space and coalesce according to some Λ-coalescent mechanism, are shown to be appropriate approximations to the genealogy of a sample of individuals. We then consider as the graph G the two dimensional torus with side length 2L+1 and show that as L tends to infinity, and time is rescaled appropriately, the partition structure of spatial Λ-coalescents of individuals sampled far enough apart converges to the partition structure of a non-spatial Kingman coalescent. From a biological point of view this means that in certain circumstances both the spatial structure as well as larger variances of the underlying offspring distribution are harder to detect from the sample. However, supplemental simulations show that for moderately large L the different structure is still evident.

A New Characterization of Endogeny

Aldous and Bandyopadhyay have shown that each solution to a recursive distributional equation (RDE) gives rise to a recursive tree process (RTP), which is a sort of Markov chain in which time has a tree-like structure and in which the state of each vertex is a random function of its descendants. If the state at the root is measurable with respect to the sigma field generated by the random functions attached to all vertices, then the RTP is said to be endogenous. For RTPs defined by continuous maps, Aldous and Bandyopadhyay showed that endogeny is equivalent to bivariate uniqueness, and they asked if the continuity hypothesis can be removed. We introduce a higher-level RDE that through its n-th moment measures contains all n-variate RDEs. We show that this higher-level RDE has minimal and maximal fixed points with respect to the convex order, and that these coincide if and only if the corresponding RTP is endogenous. As a side result, this allows us to answer the question of Aldous and Bandyopadhyay positively.

Stochastische Modelle in der Populationsgenetik

Stochastische Modelle bilden eine der wichtigsten Grundlagen fur die Datenanalyse in der Populationsgenetik sowie in der Genetik im Allgemeinen. In der Populationsgenetik geht es darum, die Vorgeschichte von Populationen uber moglicherweise sehr lange Zeitraume (Tausende von Jahren bis Jahrmillionen) anhand der in der Gegenwart zu beobachtenden genetischen Vielfalt herzuleiten. Dazu werden mathematische Modelle benotigt, welche die zu erwartende genetische Vielfalt unter bestimmten Grundannahmen wie beispielsweise einer wachsenden Population berechnen lassen. Da das Element des Zufalls bei der Weitergabe von Genen eine entscheidende Rolle spielt, handelt es sich zumeist um stochastische Modelle, in denen allen moglichen Ereignissen gewisseWahrscheinlichkeiten zugeordnet sind. Durch Vergleich mit den Beobachtungen lasst sich dann feststellen, welche Szenarien fur die vergangene Entwicklung der betrachteten Population mehr oder weniger wahrscheinlich sind.