• Corpus ID: 210839190

Optimal Gaussian concentration bounds for stochastic chains of unbounded memory

  title={Optimal Gaussian concentration bounds for stochastic chains of unbounded memory},
  author={Jean-Ren{\'e} Chazottes and Sandro Gallo and Daniel Y. Takahashi},
  journal={arXiv: Probability},
We obtain explicit and optimal Gaussian concentration bounds (GCBs) for stochastic chains of unbounded memory (SCUMs) on countable alphabets. These stochastic processes are also known as "chains with complete connections" or "g-measures". We prove that a GCB holds when the sum of oscillations of the kernel is less than one, or when the variation of the kernel is summable, i.e., belongs to l^1(N). The proof is based on maximal coupling. Our conditions are optimal in the sense that we exhibit… 

Sparse Markov Models for High-dimensional Inference

It is proved that if only few lags are relevant the authors can consistently and efficiently recover the lags and estimate the transition probabilities of high-dimensional MTD models.

Strong mixing properties of discrete-valued time series with exogenous covariates

We derive strong mixing conditions for many existing discrete-valued time series models that include exogenous covariates in the dynamic. Our main contribution is to study how a mixing condition on

of the Bernoulli Society for Mathematical Statistics and Probability Volume Twenty Six Number Four November 2020

The papers published in Bernoulli are indexed or abstracted in Current Index to Statistics, Mathematical Reviews, Statistical Theory and Method Abstracts-Zentralblatt (STMA-Z), and Zentralblatt für

Mixing rates for potentials of non-summable variations

Abstract Mixing rates, relaxation rates, and decay of correlations for dynamics defined by potentials with summable variations are well understood, but little is known for non-summable variations.

Gaussian Concentration and Uniqueness of Equilibrium States in Lattice Systems

We consider equilibrium states (that is, shift-invariant Gibbs measures) on the configuration space SZd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts}

Time series models on the simplex, with an application to dynamic modeling of relative abundance data in Ecology

A general approach to model time series on the simplex based on a general construction of infinite memory models, called chains with complete connections, which are useful to analyze abundance data in ecosystems.

Coupling and perturbation techniques for categorical time series

We present a general approach for studying autoregressive categorical time series models with dependence of infinite order and defined conditional on an exogenous covariate process. To this end, we



Attractive regular stochastic chains: perfect simulation and phase transition

Abstract We prove that uniqueness of the stationary chain, or equivalently, of the $g$-measure, compatible with an attractive regular probability kernel is equivalent to either one of the following

On Concentration Inequalities and Their Applications for Gibbs Measures in Lattice Systems

Borders on the speed of convergence of the empirical measure in the sense of Kantorovich distance, fluctuation bounds in the Shannon–McMillan–Breiman theorem, fluctuated bounds for the first occurrence of a pattern, as well as almost-sure central limit theorems are given.

Concentration inequalities for Markov processes via coupling

We obtain moment and Gaussian bounds for general coordinate-wise Lipschitz functions evaluated along the sample path of a Markov chain. We treat Markov chains on general (possibly unbounded) state

Chains with Complete Connections and One-Dimensional Gibbs Measures

The equivalence of uniqueness criteria for chains and fields is discussed, bounds for the continuity rates of the respective systems of finite-volume conditional probabilities are established and a (re)construction theorem for specifications starting from single-site conditioning is proved.


Bernoullicity is the strongest mixing property that a measuretheoretic dynamical system can have. This is known to be intimately connected to the so-called d̄ metric on processes, introduced by

Markov approximation and consistent estimation of unbounded probabilistic suffix trees

The weak consistency of a modification of Rissanen's algorithm Context which estimates the length of the suffix needed to predict the next symbol, given a finite sample is proved for infinite order chains whose transition probabilities depend on a finite suffix of the past.

Chains with unbounded variable length memory: perfect simulation and a visible regeneration scheme

  • S. Gallo
  • Computer Science, Mathematics
    Advances in Applied Probability
  • 2011
A new perfect simulation algorithm for stationary chains having unbounded variable length memory is presented, for which the family of transition probabilities is represented by a probabilistic context tree.

Concentration Inequalities - A Nonasymptotic Theory of Independence

Deep connections with isoperimetric problems are revealed whilst special attention is paid to applications to the supremum of empirical processes.

Concentration Inequalities for Dependent Random Variables via the Martingale Method

The martingale method is used to establish concentration inequalities for a class of dependent random sequences on a countable state space, with the constants in the inequalities expressed in terms