Explicit laws of large numbers for random nearest-neighbour-type graphs

Abstract

Under the unifying umbrella of a general result of Penrose & Yukich [Ann. Appl. Probab., (2003) 13, 277-303] we give laws of large numbers (in the L p sense) for the total power-weighted length of several nearest-neighbour type graphs on random point sets in R d , d ∈ N. Some of these results are known; some are new. We give limiting constants explicitly, where previously they have been evaluated in less generality or not at all. The graphs we consider include the k-nearest neighbours graph, the Gabriel graph, the minimal directed spanning forest, and the on-line nearest-neighbour graph.

Limit theory for the random on‐line nearest‐neighbor graph

“…Explicit laws of large numbers for the random ONG in (0, 1) d are given in [25]. In the present paper we give further results on the limiting behaviour in general dimensions d.…”

“…Our results for the ONG in general dimensions are as follows, and constitute a distributional convergence result for α > d, and asymptotic behaviour of the mean for α = d. For the sake of completeness, we include the law of large numbers for α < d from [25] as part (i) of the theorem below. …”

“…We use this notation to be consistent with [25], which presents explicit laws of large numbers for nearest-neighbour graphs including this one. Let U n denote the binomial point process consisting of n independent uniform random points in the unit interval.…”

“…The sum converges almost surely since it has non-negative terms and, by (25), has finite expectation for α > d. Let k ∈ N. By (25) and Hölder's inequality, there exists a constant…”

“…Also, writing · op for the operator norm, for condition (25) in [11], (74) for α > 1/2. Finally, for condition (26) in [11], for α > 0 and any ℓ ∈ N, as n → ∞…”