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Pré-Publication, Document De Travail Année : 2020

Optimal Gaussian concentration bounds for stochastic chains of unbounded memory

Résumé

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 examples of SCUMs that do not have GCB and the sum of oscillations is strictly larger than one, or the variation belongs to l^{1+epsilon) (N) for any epsilon > 0. These examples are based on the existence of phase transitions. We also extend the validity of GCB to a class of functions which can depend on infinitely many coordinates. We illustrate our results by three main applications. First, we derive a Dvoretzky-Kiefer-Wolfowitz type inequality which gives a uniform control on the fluctuations of the empirical measure. Second, in the finite-alphabet case, we apply our bounds to obtain an upper bound on the dbar-distance between two stationary SCUMs and, as a by-product, we obtain new bounds on the speed of Markovian approximation in dbar. Third, we obtain exponential rate of convergence for Birkhoff sums of a certain class of observables.
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Dates et versions

hal-02444596 , version 1 (18-01-2020)
hal-02444596 , version 2 (23-03-2020)

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  • HAL Id : hal-02444596 , version 1

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Jean-Rene Chazottes, Sandro Gallo, Daniel Takahashi. Optimal Gaussian concentration bounds for stochastic chains of unbounded memory. 2020. ⟨hal-02444596v1⟩
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