  Consider an unlimited homogeneous medium disturbed by points generated via
Poisson process. The neighborhood of a point plays an important role in spatial
statistics problems. Here, we obtain analytically the distance statistics to
$k$th nearest neighbor in a $d$-dimensional media. Next, we focus our attention
in high dimensionality and high neighborhood order limits. High dimensionality
makes distance distribution behavior as a delta sequence, with mean value equal
to Cerf's conjecture. Distance statistics in high neighborhood order converges
to a Gaussian distribution. The general distance statistics can be applied to
detect departures from Poissonian point distribution hypotheses as proposed by
Thompson and generalized here.
