This report is devoted to the forecast of the UEFA EURO 2020, Europes continental football championship, taking place across Europe in June/July 2021. We present the simulation results for this tournament, where the simulations are based on a zero-in
flated generalized Poisson regression model that includes the Elo points of the participating teams and the location of the matches as covariates and incorporates differences of team-specific skills. The proposed model allows predictions in terms of probabilities in order to quantify the chances for each team to reach a certain stage of the tournament. We use Monte Carlo simulations for estimating the outcome of each single match of the tournament, from which we are able to simulate the whole tournament itself. The model is fitted on all football games of the participating teams since 2014 weighted by date and importance.
In this article we consider transient random walks on HNN extensions of finitely generated groups. We prove that the rate of escape w.r.t. some generalised word length exists. Moreover, a central limit theorem with respect to the generalised word len
gth is derived. Finally, we show that the rate of escape, which can be regarded as a function in the finitely many parameters which describe the random walk, behaves as a real-analytic function in terms of probability measures of constant support.
We propose an approach for the analysis and prediction of a football championship. It is based on Poisson regression models that include the Elo points of the teams as covariates and incorporates differences of team-specific effects. These models for
the prediction of the FIFA World Cup 2018 are fitted on all football games on neutral ground of the participating teams since 2010. Based on the model estimates for single matches Monte-Carlo simulations are used to estimate probabilities for reaching the different stages in the FIFA World Cup 2018 for all teams. We propose two score functions for ordinal random variables that serve together with the rank probability score for the validation of our models with the results of the FIFA World Cups 2010 and 2014. All models favor Germany as the new FIFA World Champion. All possible courses of the tournament and their probabilities are visualized using a single Sankey diagram.
Random walks on a group $G$ model many natural phenomena. A random walk is defined by a probability measure $p$ on $G$. We are interested in asymptotic properties of the random walks and in particular in the linear drift and the asymptotic entropy. I
f the geometry of the group is rich, then these numbers are both positive and the way of dependence on $p$ is itself a property of $G$. In this note, we review recent results about the regularity of the drift and the entropy for free groups, free products and hyperbolic groups.
In this article we prove existence of the asymptotic entropy for isotropic random walks on regular Fuchsian buildings. Moreover, we give formulae for the asymptotic entropy, and prove that it is equal to the rate of escape of the random walk with res
pect to the Green distance. When the building arises from a Fuchsian Kac-Moody group our results imply results for random walks induced by bi-invariant measures on these groups, however our results are proven in the general setting without the assumption of any group acting on the building. The main idea is to consider the retraction of the isotropic random walk onto an apartment of the building, to prove existence of the asymptotic entropy for this retracted walk, and to `lift this in order to deduce the existence of the entropy for the random walk on the building.
The connective constant $mu(G)$ of a graph $G$ is the asymptotic growth rate of the number $sigma_{n}$ of self-avoiding walks of length $n$ in $G$ from a given vertex. We prove a formula for the connective constant for free products of quasi-transiti
ve graphs and show that $sigma_{n}sim A_{G} mu(G)^{n}$ for some constant $A_{G}$ that depends on $G$. In the case of finite products $mu(G)$ can be calculated explicitly and is shown to be an algebraic number.
We prove existence of asymptotic entropy of random walks on regular languages over a finite alphabet and we give formulas for it. Furthermore, we show that the entropy varies real-analytically in terms of probability measures of constant support, whi
ch describe the random walk. This setting applies, in particular, to random walks on virtually free groups.
We consider random walks on the set of all words over a finite alphabet such that in each step only the last two letters of the current word may be modified and only one letter may be adjoined or deleted. We assume that the transition probabilities d
epend only on the last two letters of the current word. Furthermore, we consider also the special case of random walks on free products by amalgamation of finite groups which arise in a natural way from random walks on the single factors. The aim of this paper is to compute several equivalent formulas for the rate of escape with respect to natural length functions for these random walks using different techniques.
