# Optimal Gaussian concentration bounds for stochastic chains of unbounded memory

@article{Chazottes2020OptimalGC, 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}, year={2020} }

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…

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## References

SHOWING 1-10 OF 56 REFERENCES

### Attractive regular stochastic chains: perfect simulation and phase transition

- MathematicsErgodic Theory and Dynamical Systems
- 2013

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

- Mathematics, Computer Science
- 2016

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

- Mathematics
- 2008

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

- Mathematics, Computer Science
- 2003

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.

### CRITERIA FOR D-CONTINUITY

- Mathematics
- 1998

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

- Mathematics, Computer Science
- 2006

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

- Computer Science, MathematicsAdvances 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

- MathematicsConcentration Inequalities
- 2013

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

- Mathematics
- 2008

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…