Limit theory for the random on-line nearest-neighbour graph

In the on-line nearest-neighbour graph (ONG), each point after the first in a sequence of points in Rd is joined by an edge to its nearest-neighbour amongst those points that precede it in the sequence. We study the large-sample asymptotic behaviour of the total power-weighted length of the ONG on uniform random points in $(0,1)^d$. In particular, for $d=1$ and weight exponent $\alpha>1/2$, the limiting distribution of the centred total weight is characterized by a distributional fixed-point equation. As an ancillary result, we give exact expressions for the expectation and variance of the standard nearest-neighbour (directed) graph on uniform random points in the unit interval.