Related papers: Reconstruction on Trees: Exponential Moment Bounds…

Recently introduced and studied in arXiv:2407.07888, a self-similar Markov tree (ssMt) is a random decorated tree that vastly generalises the fragmentation tree. We study here the critical case that was left aside in arXiv:2407.07888.…

We establish non-asymptotic error bounds for the classical Maximal Likelihood Estimation of the transition matrix of a given Markov chain. Meanwhile, in the reversible case, we propose a new reversibility-preserving online Symmetric…

This work is motivated by the challenges of applying the sample average approximation (SAA) method to multistage stochastic programming with an unknown continuous-state Markov process. While SAA is widely used in static and two-stage…

The Swendsen-Wang algorithm is a sophisticated, widely-used Markov chain for sampling from the Gibbs distribution for the ferromagnetic Ising and Potts models. This chain has proved difficult to analyze, due in part to the global nature of…

We consider the contact process with infection rate $\lambda$ on $\mathbb{T}_n^d$, the $d$-ary tree of height $n$. We study the extinction time $\tau_{\mathbb{T}_n^d}$, that is, the random time it takes for the infection to disappear when…

We extend Hoeffding's lemma to general-state-space and not necessarily reversible Markov chains. Let $\{X_i\}_{i \ge 1}$ be a stationary Markov chain with invariant measure $\pi$ and absolute spectral gap $1-\lambda$, where $\lambda$ is…

We consider branching processes for structured populations: each individual is characterized by a type or trait which belongs to a general measurable state space. We focus on the supercritical recurrent case, where the population may…

Reconstructing evolutionary trees from molecular sequence data is a fundamental problem in computational biology. Stochastic models of sequence evolution are closely related to spin systems that have been extensively studied in statistical…

We consider the distributional fixed-point equation: $$R \stackrel{\mathcal{D}}{=} Q \vee \left( \bigvee_{i=1}^N C_i R_i \right),$$ where the $\{R_i\}$ are i.i.d.~copies of $R$, independent of the vector $(Q, N, \{C_i\})$, where $N \in…

Markov random fields are used to model high dimensional distributions in a number of applied areas. Much recent interest has been devoted to the reconstruction of the dependency structure from independent samples from the Markov random…

On the rooted $k$-ary tree we consider a 0-1 kinetically constrained spin model in which the occupancy variable at each node is re-sampled with rate one from the Bernoulli(p) measure iff all its children are empty. For this process the…

In the study of Ising models on large locally tree-like graphs, in both rigorous and non-rigorous methods one is often led to understanding the so-called belief propagation distributional recursions and its fixed points. We prove that there…

The general Markov plus invariable sites (GM+I) model of biological sequence evolution is a two-class model in which an unknown proportion of sites are not allowed to change, while the remainder undergo substitutions according to a Markov…

The reconstruction of phylogenetic trees from molecular sequence data relies on modelling site substitutions by a Markov process, or a mixture of such processes. In general, allowing mixed processes can result in different tree topologies…

Let $X_n(k)$ be the number of vertices at level $k$ in a random recursive tree with $n+1$ vertices. We prove a functional limit theorem for the vector-valued process $(X_{[n^t]}(1),\ldots, X_{[n^t]}(k))_{t\geq 0}$, for each $k\in\mathbb N$.…

Hanson-Wright inequality provides a powerful tool for bounding the norm $|\xi|$ of a centered stochastic vector $\xi$ with sub-gaussian behavior. This paper extends the bounds to the case when $\xi$ only has bounded exponential moments of…

The invariant measure is a fundamental object in the theory of Markov processes. In finite dimensions a Markov process is defined by transition rates of the corresponding stochastic matrix. The Markov tree theorem provides an explicit…

In a complete graph $K_n$ with edge weights drawn independently from a uniform distribution $U(0,1)$ (or alternatively an exponential distribution $\operatorname{Exp}(1)$), let $T_1$ be the MST (the spanning tree of minimum weight) and let…

Let $R_T$ be the Ricci matrix of a finite tree $T$ introduced in \cite{BaiChengHua2026}, the largest eigenvalue $\lambda_{\max}(R_T)$ determines the sign of a discrete Einstein metric curvature on the tree. This paper investigates the…

Consider a tree network $T$, where each edge acts as an independent copy of a given channel $M$, and information is propagated from the root. For which $T$ and $M$ does the configuration obtained at level $n$ of $T$ typically contain…