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).