This paper develops theory for deep neural network (DNN) estimators under dependent data.
To provide theory applicable to a variety of DNN-based estimators, I first establish nonasymptotic probability bounds on the theoretical and empirical $\mathcal{L}^{2}$-errors of nonparametric sieve estimators for a general class of estimation problems under possibly nonstationary $β$-mixing data taking values in unbounded sets.
I then apply the theory to fully connected and convolutional DNN estimators without bounds or sparsity restrictions on the DNN weights.
For both DNN classes, I derive general results when the function to be estimated is Hölder smooth and the data are nonstationary, subgaussian, and $β$-mixing with either exponential or polynomial decay.
I then specialize these to nonparametric regression, logistic regression, and quantile regression settings.
Under exponential $β$-mixing, the resulting estimators attain the nonparametric minimax rate of Stone (1982) up to logarithmic factors.