  The large sample theory of estimators for density modes is well understood.
In this paper we consider density ridges, which are a higher-dimensional
extension of modes. Modes correspond to zero-dimensional, local high-density
regions in point clouds. Density ridges correspond to $s$-dimensional, local
high-density regions in point clouds. We establish three main results. First we
show that under appropriate regularity conditions, the local variation of the
estimated ridge can be approximated by an empirical process. Second, we show
that the distribution of the estimated ridge converges to a Gaussian process.
Third, we establish that the bootstrap leads to valid confidence sets for
density ridges.
