Suppose {Xi} is an alpha-mixing stochastic process assuming values in a set X, and that faX → R is bounded and measurable. It is shown in this note that the sequence of empirical means (1/m) ∑i=1mf(xi) converges in probability to the true expected value of the function f(.). Moreover, explicit estimates are constructed of the rate at which the empirical mean converges to the true expected value. These estimates generalize classical inequalities of Hoeffding, Bennett and Bernstein to the case of alpha-mixing inputs. In earlier work, similar results have been established when the alpha-mixing coefficient of the stochastic process converges to zero at a geometric rate. No such assumption is made in the present note. This result is then applied to the problem of PAC (probably approximately correct) learning under a fixed distribution.
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机译：假设{X i }是一个alpha混合随机过程，假定集合X中的值，并且faX→R是有界且可测量的。在该注释中表明，经验均值（1 / m）∑ i = 1 m f（x i ）的序列收敛于函数f（。）的真实期望值的概率。此外，对经验均值收敛到真实期望值的比率进行了显式估计。这些估计将Hoeffding，Bennett和Bernstein的经典不等式推广到alpha混合输入的情况。在较早的工作中，当随机过程的alpha混合系数以几何速率收敛到零时，已经建立了相似的结果。在本说明中未作此假设。然后将此结果应用于固定分布下的PAC学习（可能近似正确）问题。
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