  This paper focuses on learning rate analysis of Nystr\"{o}m regularization
with sequential sub-sampling for $\tau$-mixing time series. Using a recently
developed Banach-valued Bernstein inequality for $\tau$-mixing sequences and an
integral operator approach based on second-order decomposition, we succeed in
deriving almost optimal learning rates of Nystr\"{o}m regularization with
sequential sub-sampling for $\tau$-mixing time series. A series of numerical
experiments are carried out to verify our theoretical results, showing the
excellent learning performance of Nystr\"{o}m regularization with sequential
sub-sampling in learning massive time series data. All these results extend the
applicable range of Nystr\"{o}m regularization from i.i.d. samples to
non-i.i.d. sequences.
