An Inequality of the Bernstein-type for $β$-mixing Time Series

In this article we present an inequality of the Bernstein type for sums of random variables of the form $Z_k = f(X_k,X_t)$ for $k,t\in \mathbb{N}$, $t$ fixed, where the underlying process $X$ is $\beta$-mixing. The inequality can be used to derive concentration inequalities. It can be useful to obtain consistency properties for nonparametric estimators of regression functions.