Asymptotic theory for the multidimensional random on-line nearest-neighbour graph

The on-line nearest-neighbour graph on a sequence of n uniform random points in ( 0 , 1 ) d ( d ∈ N ) joins each point after the first to its nearest neighbour amongst its predecessors. For the total power-weighted edge-length of this graph, with weight exponent α ∈ ( 0 , d / 2 ] , we prove O ( max { n 1 − ( 2 α / d ) , log n } ) upper bounds on the variance. On the other hand, we give an n → ∞ large-sample convergence result for the total power-weighted edge-length when α > d / 2 . We prove corresponding results when the underlying point set is a Poisson process of intensity n .