  Local general depth ($LGD$) functions are used for describing the local
geometric features and mode(s) in multivariate distributions. In this paper, we
undertake a rigorous systematic study of $LGD$ and establish several analytical
and statistical properties. First, we show that, when the underlying
probability distribution is absolutely continuous with density $f(\cdot)$, the
scaled version of $LGD$ (referred to as $\tau$-approximation) converges,
uniformly and in $L^d(\mathbb{R}^p)$ to $f(\cdot)$ when $\tau$ converges to
zero. Second, we establish that, as the sample size diverges to infinity the
centered and scaled sample $LGD$ converge in distribution to a centered
Gaussian process uniformly in the space of bounded functions on
$\mathcal{H}_G$, a class of functions yielding $LGD$. Third, using the sample
version of the $\tau$-approximation ($S \tau A$) and the gradient system
analysis, we develop a new clustering algorithm. The validity of this algorithm
requires several results concerning the uniform finite difference approximation
of the gradient system associated with $S \tau A$. For this reason, we
establish \emph{Bernstein}-type inequality for deviations between the centered
and scaled sample $LGD$, which is also of independent interest. Finally,
invoking the above results, we establish consistency of the clustering
algorithm. Applications of the proposed methods to mode estimation and upper
level set estimation are also provided. Finite sample performance of the
methodology are evaluated using numerical experiments and data analysis.
