This dissertation studies nonparametric estimation with deep neural networks (DNNs) under dependent data. The first chapter introduces my work, describes its contribution to the literature, and defines the mathematical notation used throughout.

The second chapter establishes statistical properties of deep neural network (DNN) estimators under dependent data. Two general results for nonparametric sieve estimators directly applicable to DNN estimators are given. The first establishes rates for convergence in probability under nonstationary data. The second provides non-asymptotic probability bounds on L²-errors under stationary β-mixing data. I apply these results to DNN estimators in both regression and classification contexts imposing only a standard Hölder smoothness assumption. The DNN architectures considered are common in applications, featuring fully connected feedforward networks with any continuous piecewise linear activation function, unbounded weights, and a width and depth that grows with sample size. The framework provided also offers potential for research into other DNN architectures and time-series applications.

The third chapter demonstrates the practical implications of these results in a partially linear regression model under stationary β-mixing data. In this setting, I obtain √n-asymptotic normality on the finite dimensional parameter after first-stage DNN estimation of infinite dimensional parameters.