GAUSSIAN CONCENTRATION BOUNDS FOR STOCHASTIC CHAINS OF UNBOUNDED MEMORY

• 成果类型: Article
• 署名作者: Chazottes, Jean-Rene; Gallo, Sandro; Takahashi, Daniel Y.
• 署名单位: Institut Polytechnique de Paris; Ecole Polytechnique; Centre National de la Recherche Scientifique (CNRS); Universidade Federal de Sao Carlos; Universidade Federal do Rio Grande do Norte
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/22-AAP1893
• 发表日期: 2023
• 页码: 3321-3350
• 关键词: complete connections; Concentration inequalities; perfect simulation; markov approximations; random-variables; uniqueness; nonuniqueness; divergence; FIELDS

摘要

Stochastic chains of unbounded memory (SCUMs) are generalization of Markov chains, also known in the literature as chains with complete con-nections or g-measures. We obtain Gaussian concentration bounds (GCB) in this large class of models, for general alphabets, under two different con-ditions on the kernel: (1) when the sum of its oscillations is less than one, or (2) when the sum of its variations is finite, that is, belongs to l(1)(N). We also obtain explicit constants as functions of the parameters of the model. Our conditions are sharp in the sense that we exhibit examples of SCUMs that do not have GCB and for which the sum of oscillations is 1 + is an element of, or the variation belongs to l(1+is an element of)(N) for any e > 0. These examples are based on the existence of phase transitions.We illustrate our results with four 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 obtain an upper bound on the d -distance between two stationary SCUMs and, as a by-product, we obtain new explicit bounds on the speed of Markovian approximation in d . Third, we derive new bounds on the fluctu-ations of the plug-in estimator for entropy. Fourth, we obtain new rate of convergence for the maximum likelihood estimator of conditional probability.