  We consider the sparse estimation for stochastic processes with possibly
infinite-dimensional nuisance parameters, by using the Dantzig selector which
is a sparse estimation method similar to $Z$-estimation. When a consistent
estimator for a nuisance parameter is obtained, it is possible to construct an
asymptotically normal estimator for the parameter of interest under appropriate
conditions. Motivated by this fact, we establish the asymptotic behavior of the
Dantzig selector for models of ergodic stochastic processes with
high-dimensional parameters of interest and possibly infinite-dimensional
nuisance parameters. Applications to ergodic time series models including
integer-valued autoregressive models and ergodic diffusion processes are
presented.
