  Empirical process theory for i.i.d. observations has emerged as a ubiquitous
tool for understanding the generalization properties of various statistical
problems. However, in many applications where the data exhibit temporal
dependencies (e.g., in finance, medical imaging, weather forecasting etc.), the
corresponding empirical processes are much less understood. Motivated by this
observation, we present a general bound on the expected supremum of empirical
processes under standard $\beta/\rho$-mixing assumptions. Unlike most prior
work, our results cover both the long and the short-range regimes of
dependence. Our main result shows that a non-trivial trade-off between the
complexity of the underlying function class and the dependence among the
observations characterizes the learning rate in a large class of nonparametric
problems. This trade-off reveals a new phenomenon, namely that even under
long-range dependence, it is possible to attain the same rates as in the i.i.d.
setting, provided the underlying function class is complex enough. We
demonstrate the practical implications of our findings by analyzing various
statistical estimators in both fixed and growing dimensions. Our main examples
include a comprehensive case study of generalization error bounds in
nonparametric regression over smoothness classes in fixed as well as growing
dimension using neural nets, shape-restricted multivariate convex regression,
estimating the optimal transport (Wasserstein) distance between two probability
distributions, and classification under the Mammen-Tsybakov margin condition --
all under appropriate mixing assumptions. In the process, we also develop
bounds on $L_r$ ($1\le r\le 2$)-localized empirical processes with dependent
observations, which we then leverage to get faster rates for (a) tuning-free
adaptation, and (b) set-structured learning problems.
