arXiv subjects

Stein Andreas Bethuelsen

Publications and source records attributed to Stein Andreas Bethuelsen.

17 recordsLinked to original sources

arXiv2026

Multi-regime Markov-switching models with time-varying transition probabilities: An application to U.S. Treasury yields

This paper studies Markov-switching (MS) models with time-varying transition probabilities (TVTP) under alternative specifications of the transition probability matrix. We extend the two-regime common-variance setting of the Generalized Autoregressive Score (GAS) model of Bazzi et al. (2017) to the general $K$-regime case with regime-specific means and variances, and develop an open-source R package, multiregimeTVTP, for simulation and estimation of such models. A Monte Carlo study shows that the regime means, variances, and transition probabilities are reliably recovered, whereas the TVTP driving coefficients are harder to identify. The GAS score coefficient appears to be statistically non-identifiable, owing to a ridge in the likelihood surface linking it to the regime variance. The filtered regime probabilities are accurately recovered under correct specification, whereas the one-step-ahead conditional mean is insensitive to misspecification of the transition dynamics. An empirical application to U.S. Treasury zero-coupon yield changes (1961-2024) at four maturities shows that a specification driven by the lagged yield level provides the best fit, attaining the lowest AIC at all four maturities and the lowest BIC at the short end of the yield curve, and that the estimated regimes align with documented episodes of U.S. monetary history.

stat.ME↗

arXiv2026

The Zombie Infection Model

We study a variant of the stochastic SIR model on graphs that has previously been introduced in the physics literature for modelling zombie outbreaks and here referred to as the Zombie Infection Model (ZIM). In this model, initially each node of a graph is either susceptible, infected or removed. As in the SIR model, a susceptible node becomes infected at rate $λ$ times the number of its infected neighbours. Moreover, in the ZIM, an infected node is removed at rate $1$ times the number of its susceptible neighbours. This process exhibits rich and sometimes counterintuitive behaviour. By combining various coupling techniques, we provide a rigorous mathematical analysis of the model, focusing on monotonicity properties and the probability of the infection spreading indefinitely. One of our main results is that this probability is monotone with respect to an increase of $λ$ for the process on trees, but that there are graphs of bounded degree for which it is continuous and yet not monotone. We also establish bounds on this probability for the process on general graphs, and derive more precise results for complete graphs, regular trees, and the $d$-dimensional integer lattice.

math.PR↗

arXiv2026

The truncation property and continuity for the long-range contact process on $\mathbb{Z}^d$

We consider a general class of contact processes on $\mathbb{Z}^d$ with potentially long-range interactions. By adapting well established renormalization arguments to the long-range setting we extend by now classical results for finite-range processes to this more general setting. Particularly, we provide general conditions on the decay of the interactions under which a supercritical process remains supercritical after truncation of the interaction parameter at a sufficiently large distance. Further, for the family of parameters satisfying this latter truncation property, we conclude that the probability of the process to never recover is continuous.

math.PR↗

arXiv2026

Quenched local limit theorem for a directed random walk on the backbone of a supercritical oriented percolation cluster for $d \ge 1$

In this work we extend the quenched local limit theorem obtained by the authors in [BBDS23]. More precisely, we consider a directed random walk on the backbone of the supercritical oriented percolation cluster in dimensions $d+1$ with $d\geq 1$ being the spatial dimension. In [BBDS23] an annealed local central limit theorem was proven for all $d\geq 1$ and a quenched local limit theorem under the assumption $d\geq 3$. Here we show that the latter result also holds for all $d \ge 1$.

math.PR↗

arXiv2025

Strict monotonicity of critical points in independent long-range percolation models

We consider independent long-range percolation models on locally finite vertex-transitive graphs. Using coupling ideas we prove strict monotonicity of the critical points with respect to local perturbations in the connection function, thereby improving upon previous results obtained via the classical essential enhancement method of Aizenman and Grimmett in several ways. In particular, our approach allows us to work under minimal assumptions, namely shift-invariance and summability of the connection function, and it applies to both undirected and directed bond percolation models.

math.PR↗

arXiv2025

Mixing for Poisson representable processes and consequences for the Ising model and the contact process

Forsström et al. [8] recently introduced a large class of $\{0,1\}$-valued processes that they named Poisson representable. In addition to deriving several interesting properties for these processes, their main focus was determining which processes are contained in this class. In this paper, we derive new characteristics for Poisson representable processes in terms of certain mixing properties. Using these, we argue that neither the upper invariant measure of the supercritical contact process on $\mathbb{Z}^d$ nor the plus state of the Ising model on $\mathbb{Z}^2$ within the phase transition regime is Poisson representable. Moreover, we show that on $\mathbb{Z}^d$, $d\geq 2$, any non-extremal translation invariant state of the Ising model cannot be Poisson representable. Together, these results provide answers to questions raised in [8].

math.PR↗

arXiv2024

Random walks on random walks: non-perturbative results in high dimensions

Consider the dynamic environment governed by a Poissonian field of independent particles evolving as simple random walks on $\mathbb{Z}^d$. The random walk on random walks model refers to a particular stochastic process on $\mathbb{Z}^d$ whose evolution at time $t$ depends on the number of such particles at its location. We derive classical limit theorems for this instrumental model of a random walk in a dynamic random environment, applicable in sufficiently high dimensions. More precisely, for $d \geq 5$, we prove a strong law of large numbers and large deviation estimates. Further, for $d\geq 9$, we obtain a functional central limit theorem under the annealed law. These results are non-perturbative in the sense that they hold for any positive density of the Poissonian field. Under the aforementioned assumptions on the dimension they therefore improve on previous work on the model. Moreover, they stand in contrast to the anomalous behaviour predicted in low dimensions.

math.PR↗

arXiv2024

Limit laws for random walks in a dynamic path-cone mixing random environment

We study the asymptotic behaviour of a random walk whose evolution is dependent on the state of an itself dynamically evolving environment. In particular, we extend our previous results in [Bethuelsen and Völlering, 2016] and prove a strong law of large numbers and large deviation estimates assuming that the dynamic environment is "path-cone"-mixing. Under a mild assumption on the decay rate of this mixing property we further obtain a functional central limit theorem under the annealed law. Our method of proofs rest on the study of the so-called local environment process and general results for $ϕ$-mixing stochastic processes.

math.PR↗

arXiv2021

Local limit theorems for a directed random walk on the backbone of a supercritical oriented percolation cluster

We consider a directed random walk on the backbone of the supercritical oriented percolation cluster in dimensions $d+1$ with $d \ge 3$ being the spatial dimension. For this random walk we prove an annealed local central limit theorem and a quenched local limit theorem. The latter shows that the quenched transition probabilities of the random walk converge to the annealed transition probabilities reweighted by a function of the medium centred at the target site. This function is the density of the unique measure which is invariant for the point of view of the particle, is absolutely continuous with respect to the annealed measure and satisfies certain concentration properties.

math.PR↗

arXiv2021

Graph constructions for the contact process with a prescribed critical rate

We construct graphs (trees of bounded degree) on which the contact process has critical rate (which will be the same for both global and local survival) equal to any prescribed value between zero and $λ_c(\mathbb{Z})$, the critical rate of the one-dimensional contact process. We exhibit both graphs in which the process at this target critical value survives (locally) and graphs where it dies out (globally).

math.PR↗

arXiv2020

Invariance principle for random walks on dynamically averaging random conductances

We prove an invariance principle for continuous-time random walks in a dynamically averaging environment on $\mathbb Z$. In the beginning, the conductances may fluctuate substantially, but we assume that as time proceeds, the fluctuations decrease according to a typical diffusive scaling and eventually approach constant unit conductances. The proof relies on a coupling with the standard continuous time simple random walk.

math.PR↗

arXiv2018

On projections of the supercritical contact process: uniform mixing and cutoff phenomenon

We consider the contact process on a countable-infinite and connected graph of bounded degree. For this process started from the upper invariant measure, we prove certain uniform mixing properties under the assumption that the infection parameter is sufficiently large. In particular, we show that the projection of such a process onto a finite subset forms a process which is $ϕ$-mixing. The proof of this is based on large deviation estimates for the spread of an infection and general correlation inequalities. In the special case of the contact process on $\mathbb{Z}^d$, $d\geq1$, we furthermore prove the cutoff phenomenon, valid in the entire supercritical regime.

math.PR↗

arXiv2018

One-sided continuity properties for the Schonmann projection

We consider the plus-phase of the two-dimensional Ising model below the critical temperature. In $1989$ Schonmann proved that the projection of this measure onto a one-dimensional line is not a Gibbs measure. After many years of continued research which have revealed further properties of this measure, the question whether or not it is a Gibbs measure in an almost sure sense remains open. In this paper we study the same measure by interpreting it as a temporal process. One of our main results is that the Schonmann projection is almost surely a regular $g$-measure. That is, it does possess the corresponding one-sided notion of almost Gibbsianness. We further deduce strong one-sided mixing properties which are of independent interest. Our proofs make use of classical coupling techniques and some monotonicity properties which are known to hold for one-sided, but not two-sided conditioning for FKG measures.

math.PR↗

arXiv2017

Stochastic Domination in Space-Time for the Contact Process

Liggett and Steif (2006) proved that, for the supercritical contact process on certain graphs, the upper invariant measure stochastically dominates an i.i.d.\ Bernoulli product measure. In particular, they proved this for $\mathbb{Z}^d$ and (for infection rate sufficiently large) $d$-ary homogeneous trees $T_d$. In this paper we prove some space-time versions of their results. We do this by combining their methods with specific properties of the contact process and general correlation inequalities. One of our main results concerns the contact process on $T_d$ with $d\geq2$. We show that, for large infection rate, there exists a subset $Δ$ of the vertices of $T_d$, containing a "positive fraction" of all the vertices of $T_d$, such that the following holds: The contact process on $T_d$ observed on $Δ$ stochastically dominates an independent spin-flip process. (This is known to be false for the contact process on graphs having subexponential growth.) We further prove that the supercritical contact process on $\mathbb{Z}^d$ observed on certain $d$-dimensional space-time slabs stochastically dominates an i.i.d.\ Bernoulli product measure, from which we conclude strong mixing properties important in the study of certain random walks in random environment.

math.PR↗

arXiv2016

Absolute Continuity and Weak Uniform Mixing of Random Walk in Dynamic Random Environment

We prove results for random walks in dynamic random environments which do not require the strong uniform mixing assumptions present in the literature. We focus on the "environment seen from the walker"-process and in particular its invariant law. Under general conditions it exists and is mutually absolutely continuous to the environment law. With stronger assumptions we obtain for example uniform control on the density or a quenched CLT. The general conditions are made more explicit by looking at hidden Markov models or Markov chains as environment and by providing simple examples.

math.PR↗

arXiv2016

Law of large numbers for random walks on attractive spin-flip dynamics

We prove a law of large numbers for certain random walks on certain attractive dynamic random environments when initialised from all sites equal to the same state. This result applies to random walks on $\mathbb{Z}^d$ with $d\geq1$. We further provide sufficient mixing conditions under which the assumption on the initial state can be relaxed, and obtain estimates on the large deviation behaviour of the random walk. As prime example we study the random walk on the contact process, for which we obtain a law of large numbers in arbitrary dimension. For this model, further properties about the speed are derived.

math.PR↗

arXiv2016

The contact process as seen from a random walk

We consider a random walk on top of the contact process on $\mathbb{Z}^d$ with $d\geq 1$. In particular, we focus on the "contact process as seen from the random walk". Under the assumption that the infection rate of the contact process is large or the jump rate of the random walk is small, we show that this process has at most two extremal measures. Moreover, the convergence to these extremal measures is characterised by whether the contact process survives or dies out, similar to the complete convergence theorem known for the ordinary contact process. Using this, we furthermore provide a law of large numbers for the random walk which holds under general assumptions on the jump probabilities of the random walk.

math.PR↗