  Deep neural networks are powerful tools to model observations over time with
non-linear patterns. Despite the widespread use of neural networks in such
settings, most theoretical developments of deep neural networks are under the
assumption of independent observations, and theoretical results for temporally
dependent observations are scarce. To bridge this gap, we study theoretical
properties of deep neural networks on modeling non-linear time series data.
Specifically, non-asymptotic bounds for prediction error of (sparse)
feed-forward neural network with ReLU activation function is established under
mixing-type assumptions. These assumptions are mild such that they include a
wide range of time series models including auto-regressive models. Compared to
independent observations, established convergence rates have additional
logarithmic factors to compensate for additional complexity due to dependence
among data points. The theoretical results are supported via various numerical
simulation settings as well as an application to a macroeconomic data set.
