Central limit theorems for k-nearest neighbour distances

Let X1,X2,X3,… be independent d-dimensional variables with common density function f. Let Ri,k,n be the distance from Xi to its kth nearest neighbour in {X1,…,Xn}. Suppose (kn) is a sequence with 1⪡kn⪡n2/(2+d) as n tends to infinity (or 1⪡kn⪡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,nd, 1⩽i⩽n.