Let \(\{(Y_t,X_t)\}_{t\in \mathbb {Z}}\) be a strictly stationary process with \(Y_t=f(X_t)+\xi _t\), where \(f:[0,1]^d\to \mathbb {R}\) is the unknown regression function and \(\xi _t\) is the noise. In regression analysis, one wants to estimate the regression function from a data set \(\{(Y_i,X_i)\}_{i=1}^n\) drawn from \(\{(Y_t,X_t)\}_{t\in \mathbb {Z}}\). In many applications, it is necessary to consider that the data set \(\{(Y_i,X_i)\}_{i=1}^n\) is dependent. Many kinds of mixing dependences as the \(\alpha \)-mixing dependence and the \(\beta \)-mixing dependence are considered. In this paper, when the data set \(\{(Y_i,X_i)\}_{i=1}^n\) is the \(\alpha \)-mixing dependence, \(\xi _1\) is not necessarily bounded and its distribution is not necessarily known, and the regression function is included in Besov balls, the fast rate of convergence of the adaptive wavelet estimators are given, i.e., the construction of the estimators does not depend on the unknown distribution.