CENTRAL LIMIT THEOREMS FOR K-NEAREST NEIGHBOUR DISTANCES
By Mathew D. Penrose
Let X1, X2, X3, ... be indepedent d-dimensional variables
with common density function f. Let
Ri,k,n be the distance
from Xi to its k-th nearest neighbour in
{X1, ... ,Xn}.
Suppose (kn) is a sequence with
1 << kn << n2/(2+d) as
n tends to infinity (or 1 << k_n << n2/3
for a uniform distribution).
Subject to conditions on f, we find a central limit theorem
(in the large-n limit) for a time-change of the counting
process with jumps
at the points
f(Xi) (Ri,kn,n)d,
1 <= i <= n.
Stochastic Processes and Their Applications, 85,
295-320 (2000).