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Lorenz A. Gilch

Publications and source records attributed to Lorenz A. Gilch.

At least 19 recordsLinked to original sources
arXiv2026

Central Limit Theorems for Drift and Entropy of Random Walks on Free Products

In this article we consider a natural class of random walks on free products of graphs, which arise as convex combinations of random walks on the single factors. From the works of Gilch [1,2] it is well-known that for these random walks the asymptotic entropy as well as the drift w.r.t. the natural transition graph distance and also w.r.t. the word length exist. The aim of this article is to formulate three central limit theorems with respect to both drift definitions and also w.r.t. the entropy when seen as the limit of last visit generating functions. In the case that the random walk depends on finitely many parameters we show that the corresponding variances in the central limit theorems w.r.t. both drifts vary real-analytically in terms of these parameters, while the variance in the central limit theorem w.r.t. the entropy varies real-analytically in the case of free products of finite graphs.

math.PR↗
arXiv2026

Cone-Additive Functions for Random Walks on Free Products of Graphs

We define cone-additive functions for random walks on free products of countable graphs. These functions satisfy a limit theorem under mild assumptions. In fact, cone-additivity is present in several well-studied notions, like entropy, asymptotic range and drift. Cone-additivity can be seen as a separation property by space -- a quite different perspective than the well-studied concept of sub-additivity in the context of free products of groups, which is a separation by time. In our inhomogeneous setting of free products of graphs, this separation by space allows us to deduce new limit theorems for travelling salesman problems (that is, distance functions of lamplighter random walks on free products), for weight functions on edges and the range of the $r$-th visit.

math.PR↗
arXiv2024

Asymptotic Capacity of the Range of Random Walks on Free Products of Graphs

In this article we prove existence of the asymptotic capacity of the range of random walks on free products of graphs. In particular, we will show that the asymptotic capacity of the range is almost surely constant and strictly positive. Furthermore, we provide a central limit theorem for the capacity of the range and show that it varies real-analytically in terms of finitely supported probability measures of constant support.

math.PR↗
arXiv2022

Range of Random Walks on Free Products

In this article we consider transient random walks on free products of graphs. We prove that the asymptotic range of these random walks exists and is strictly positive. In particular, we show that the range varies real-analytically in terms of probability measures of constant support. Moreover, we prove a central limit theorem associated with the range of the random walk.

math.PR↗
arXiv2022

Nested Zero Inflated Generalized Poisson Regression for FIFA World Cup 2022

This article is devoted to the forecast of the FIFA World Cup 2022 via nested zero-inflated generalized Poisson regression. Our regression model incorporates the Elo points of the participating teams, the location of the matches and the of team-specific skills in attack and defense as covariates. 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 2016 weighted by date and importance. Validation with previous tournaments and comparison with other Poisson models are given.

stat.AP↗
arXiv2021

A Data-Analytic Evaluation of Smoking Cessation via Online Hypnosis

In this article we report the observations of our field study which investigates the success rates of online hypnosis for smoking cessation. Following an international call in Germany, Austria and Switzerland for subjects willing to stop smoking $83$ participants contributed to this study, which took place during the lockdowns of the COVID-19 pandemic. All subjects received two online hypnosis sessions in small groups and additionally got two MP3 audio files for invididual intensification. The participants sent back several questionnaires, from which we present the evaluation results, and we discuss the problems observed during the study. It turned out that $86\%$ of the participants are smoke-free after $6$ months as long as they participated according to the intervention protocol. Among all smokers after six months we have observed a reduction of $32\%$ in cigarette consumption. This report shall be seen as a statistical evaluation of the collected data and not as the report of a fully pre-designed study in a narrower sense.

stat.AP↗
arXiv2021

UEFA EURO 2020 Forecast via Nested Zero-Inflated Generalized Poisson Regression

This report is devoted to the forecast of the UEFA EURO 2020, Europe's 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-inflated 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.

stat.AP↗
arXiv2020

Asymptotic Word Length of Random Walks on HNN Extensions

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 length 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.

math.PR↗
arXiv2019

Prediction Model for the Africa Cup of Nations 2019 via Nested Poisson Regression

This article is devoted to the forecast of the Africa Cup of Nations 2019 football tournament. It is based on a Poisson regression model that includes the Elo points of the participating teams 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. Monte Carlo simulations are used to estimate the outcome of each single match of the tournament and hence to simulate the whole tournament itself. The model is fitted on all football games on neutral ground of the participating teams since 2010.

stat.AP↗
arXiv2018

On Elo based prediction models for the FIFA Worldcup 2018

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.

stat.AP↗
arXiv2016

Asymptotic entropy of random walks on Fuchsian buildings and Kac-Moody 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 respect 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.

math.PR↗
arXiv2015

Regularity of the drift and entropy of random walks on groups

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. If 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.

math.PR↗
arXiv2015

Counting self-avoiding walks on free products of graphs

The connective constant $μ(G)$ of a graph $G$ is the asymptotic growth rate of the number $σ_{n}$ 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-transitive graphs and show that $σ_{n}\sim A_{G} μ(G)^{n}$ for some constant $A_{G}$ that depends on $G$. In the case of finite products $μ(G)$ can be calculated explicitly and is shown to be an algebraic number.

math.CO↗
arXiv2015

Asymptotic Entropy of Random Walks on Regular Languages over a Finite Alphabet

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, which describe the random walk. This setting applies, in particular, to random walks on virtually free groups.

math.PR↗
arXiv2011

Branching Random Walks on Free Products of Groups

We study certain phase transitions of branching random walks (BRW) on Cayley graphs of free products. The aim of this paper is to compare the size and structural properties of the trace, i.e., the subgraph that consists of all edges and vertices that were visited by some particle, with those of the original Cayley graph. We investigate the phase when the growth parameter $λ$ is small enough such that the process survives but the trace is not the original graph. A first result is that the box-counting dimension of the boundary of the trace exists, is almost surely constant and equals the Hausdorff dimension which we denote by $Φ(λ)$. The main result states that the function $Φ(λ)$ has only one point of discontinuity which is at $λ_{c}=R$ where $R$ is the radius of convergence of the Green function of the underlying random walk. Furthermore, $Φ(R)$ is bounded by one half the Hausdorff dimension of the boundary of the original Cayley graph and the behaviour of $Φ(R)-Φ(λ)$ as $λ\uparrow R$ is classified. In the case of free products of infinite groups the end-boundary can be decomposed into words of finite and words of infinite length. We prove the existence of a phase transition such that if $λ\leq \tildeλ_{c}$ the end boundary of the trace consists only of infinite words and if $λ>\tildeλ_{c}$ it also contains finite words. In the last case, the Hausdorff dimension of the set of ends (of the trace and the original graph) induced by finite words is strictly smaller than the one of the ends induced by infinite words.

math.PR↗
arXiv2011

Phase Transitions for Random Walk Asymptotics on Free Products of Groups

Suppose we are given finitely generated groups $Γ_1,...,Γ_m$ equipped with irreducible random walks. Thereby we assume that the expansions of the corresponding Green functions at their radii of convergence contain only logarithmic or algebraic terms as singular terms up to sufficiently large order (except for some degenerate cases). We consider transient random walks on the free product {$Γ_1 \ast ... \astΓ_m$} and give a complete classification of the possible asymptotic behaviour of the corresponding $n$-step return probabilities. They either inherit a law of the form $\varrho^{nδ} n^{-λ_i} \log^{κ_i}n$ from one of the free factors $Γ_i$ or obey a $\varrho^{nδ} n^{-3/2}$-law, where $\varrho<1$ is the corresponding spectral radius and $δ$ is the period of the random walk. In addition, we determine the full range of the asymptotic behaviour in the case of nearest neighbour random walks on free products of the form $\Z^{d_1}\ast ... \ast \Z^{d_m}$. Moreover, we characterize the possible phase transitions of the non-exponential types $n^{-λ_i}\log^{κ_i}n$ in the case $Γ_1\astΓ_2$.

math.PR↗
arXiv2010

Asymptotic Entropy of Random Walks on Free Products

Suppose we are given the free product V of a finite family of finite or countable sets. We consider a transient random walk on the free product arising naturally from a convex combination of random walks on the free factors. We prove the existence of the asymptotic entropy and present three different, equivalent formulas, which are derived by three different techniques. In particular, we will show that the entropy is the rate of escape with respect to the Greenian metric. Moreover, we link asymptotic entropy with the rate of escape and volume growth resulting in two inequalities.

math.PR↗