  We consider the Voronoi diagram generated by $n$ i.i.d.
$\mathbb{R}^{d}$-valued random variables with an arbitrary underlying
probability density function $f$ on $\mathbb{R}^{d}$, and analyse the
asymptotic behaviours of certain geometric properties, such as the measure, of
the Voronoi cells as $n$ tends to infinity. We adapt the methods used by
Devroye et al (2017) to conduct a study of the asymptotic properties of two
types of Voronoi cells: 1, Voronoi cells that have a fixed nucleus; 2, Voronoi
cells that contain a fixed point. For the first type of Voronoi cells, we show
that their geometric properties resemble those in the case when the Voronoi
diagram is generated by a homogeneous Poisson point process. For the second
type of Voronoi cells, we determine the limiting distribution, which is
universal in all choices of $f$, of the rescaled measure of the cells. For both
types, we establish the asymptotic independence of measures of disjoint cells.
