Related papers: A non constructive proof of the existence of a max…

We establish the weak large deviations principle for empirical measures of Markov chains on $\mathbb R^d$ under mild assumptions. In particular, no irreducibility is assumed and the initial measure may be arbitrary. The proof is entirely…

For a class of irreducible Markov chains with an infinitely countable set of states, we establish a new verifiable necessary and sufficient condition for recurrence and transience. We show that if one of the basic assumptions is not…

Using ideas borrowed from topological dynamics and ergodic theory we introduce topological and metric versions of the recurrence property for general Markov chains. The main question of interest here is how large is the set of recurrent…

We prove an invariance principle (functional central limit theorem) for a vector-valued additive functional of a Markov chain for almost every starting point with respect to an ergodic equilibrium distribution. The hypothesis is a moment…

We prove that if $\Sigma_{\mathbf A}(\mathbb N)$ is an irreducible Markov shift space over $\mathbb N$ and $f:\Sigma_{\mathbf A}(\mathbb N) \rightarrow \mathbb R$ is coercive with bounded variation then there exists a maximizing probability…

We derive explicit upper bounds for the $\bar{d}$-distance between a chain of infinite order and its canonical $k$-steps Markov approximation. Our proof is entirely constructive and involves a "coupling from the past" argument. The new…

In this note, the Zorn lemma is extended to arbitrary binary relations and thus the Zorn lemma can do for optimization when the transitivity is broken. Zorn's extended lemma can be used to prove existence theorems of generalized solution…

We consider topological Markov chains (also called Markov shifts) on countable graphs. We show that a transient graph can be extended to a recurrent graph of equal entropy which is either positive recurrent of null recurrent, and we give an…

Given a strongly stationary Markov chain and a finite set of stopping rules, we prove the existence of a polynomial algorithm which projects the Markov chain onto a minimal Markov chain without redundant information. Markov complexity is…

We obtain explicit upper bounds for the number of irreducible factors for a class of compositions of polynomials in several variables over a given field. In particular, some irreducibility criteria are given for this class of compositions…

This paper presents an elementary proof of stochastic stability of a discrete-time reversible Markov chain starting from a Foster-Lyapunov drift condition. Besides its relative simplicity, there are two salient features of the proof: (i) it…

We characterise when the log arc scheme of a fine log scheme $(X, M)$, with $X$ a variety over a field of characteristic zero, is irreducible. This generalises the theorem of Kolchin that the (ordinary) arc scheme of $X$ is irreducible…

We establish a new Bernstein-type deviation inequality for general (non-reversible) discrete-time Markov chains via an elementary approach. More robust than existing works in the literature, our result only requires the Markov chain to…

Mostof the existing literature on supervised machine learning problems focuses on the case when the training data set is drawn from an i.i.d. sample. However, many practical problems are characterized by temporal dependence and strong…

We prove that the absolute spectral gap of any monotone Markov chain coincides with its optimal Ollivier-Ricci curvature, where the word `optimal' refers to the choice of the underlying metric. Moreover, we provide a new expression in terms…

We study a class of dynamical systems generated by random substitutions, which contains both intrinsically ergodic systems and instances with several measures of maximal entropy. In this class, we show that the measures of maximal entropy…

We give an analogy between non-reversible Markov chains and electric networks much in the flavour of the classical reversible results originating from Kakutani, and later Kem\'eny-Snell-Knapp and Kelly. Non-reversibility is made possible by…

We provide two new constructions of Markov chains which had previously arisen from the representation theory of the infinite-dimensional unitary group. The first construction uses the combinatorial rule for the Littlewood-Richardson…

We give the criterion for the irreducibility, the Schur irreducibility and the indecomposability of the set of two $n\times n$ matrices $\Lambda_n$ and $A_n$ in terms of the subalgebra associated with the "support" of the matrix $A_n$,…

A maxitive measure is the analogue of a finitely additive measure or charge, in which the usual addition is replaced by the supremum operation. Contrarily to charges, maxitive measures often have a density. We show that maxitive measures…