In this paper we present a series of results that permit to extend in a direct manner uniform deviation inequalities of the empirical process from the independent to the dependent case characterizing the additional error in terms of $\beta-$mixing coefficients associated to the training sample. We then apply these results to some previously obtained inequalities for independent samples associated to the deviation of the least-squared error in nonparametric regression to derive corresponding generalization bounds for regression schemes in which the training sample may not be independent. These results provide a framework to analyze the error associated to regression schemes whose training sample comes from a large class of $\beta-$mixing sequences, including geometrically ergodic Markov samples, using only the independent case. More generally, they permit a meaningful extension of the Vapnik-Chervonenkis and similar theories for independent training samples to this class of $\beta-$mixing samples.

翻译：在本文中,我们提出了一系列结果,这些结果可以直接地将经验过程从独立体到独立体的经验过程的不平等统一偏差扩大至独立体的情况,这些结果说明与培训样本相关的额外差错是$\beta-$混合系数。然后,我们将这些结果应用于与非参数回归中最小偏差相关的独立样本先前获得的一些不平等,以便得出相应的回归回归方案的一般界限,而培训样本可能并不独立。这些结果提供了一个框架,用以分析与回归方案相关的错误,因为培训样本来自大类的$\beta-$混合序列,包括按几何测量的ergodic Markov样本,仅使用独立体。更一般地说,这些结果允许将Vapnik-Chervonenkis和关于独立培训样本的类似理论有意义地延伸至这一类的 $\beta-$混合体样本。