  We consider the sparse principal component analysis for high-dimensional
stationary processes. The standard principal component analysis performs poorly
when the dimension of the process is large. We establish the oracle
inequalities for penalized principal component estimators for the processes
including heavy-tailed time series. The rate of convergence of the estimators
is established. We also elucidate the theoretical rate for choosing the tuning
parameter in penalized estimators. The performance of the sparse principal
component analysis is demonstrated by numerical simulations. The utility of the
sparse principal component analysis for time series data is exemplified by the
application to average temperature data.
