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CENTRAL LIMIT THEOREMS FOR K-NEAREST NEIGHBOUR DISTANCES

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By Mathew D. Penrose

Let *X*_{1}, X_{2}, X_{3}, ... be indepedent *d*-dimensional variables
with common density function *f*. Let
*R*_{i,k,n} be the distance
from *X*_{i} to its *k*-th nearest neighbour in
{*X*_{1}, ... ,X_{n}}.
Suppose *(k*_{n}) is a sequence with
*1 << k*_{n} << n^{2/(2+d)} as
*n* tends to infinity (or * 1 << k_*_{n} << n^{2/3}
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) (R*_{i,kn,n})^{d},
*1 <= i <= n*.
Stochastic Processes and Their Applications, 85,
295-320 (2000).