arXiv subjects

Andrew R. Wade

Publications and source records attributed to Andrew R. Wade.

44 records · Page 3Linked to original sources

arXiv2008

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 $(0,1)^d$ ($d \in \N$) 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 $α\in (0,d/2]$, we prove $O(\max \{n^{1-(2α/d)}, \log n \})$ upper bounds on the variance. On the other hand, we give an $n \to \infty$ 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 $n$.

math.PR↗

arXiv2008

Asymptotic behaviour of randomly reflecting billiards in unbounded tubular domains

We study stochastic billiards in infinite planar domains with curvilinear boundaries: that is, piecewise deterministic motion with randomness introduced via random reflections at the domain boundary. Physical motivation for the process originates with ideal gas models in the Knudsen regime, with particles reflecting off microscopically rough surfaces. We classify the process into recurrent and transient cases. We also give almost-sure results on the long-term behaviour of the location of the particle, including a super-diffusive rate of escape in the transient case. A key step in obtaining our results is to relate our process to an instance of a one-dimensional stochastic process with asymptotically zero drift, for which we prove some new almost-sure bounds of independent interest. We obtain some of these bounds via an application of general semimartingale criteria, also of some independent interest.

math.PR↗

arXiv2007

Multivariate normal approximation in geometric probability

Consider a measure $μ_λ= \sum_x ξ_x δ_x$ where the sum is over points $x$ of a Poisson point process of intensity $λ$ on a bounded region in $d$-space, and $ξ_x$ is a functional determined by the Poisson points near to $x$, i.e. satisfying an exponential stabilization condition, along with a moments condition (examples include statistics for proximity graphs, germ-grain models and random sequential deposition models). A known general result says the $μ_λ$-measures (suitably scaled and centred) of disjoint sets in $R^d$ are asymptotically independent normals as $λ\to \infty$; here we give an $O(λ^{-1/(2d + ε)})$ bound on the rate of convergence. We illustrate our result with an explicit multivariate central limit theorem for the nearest-neighbour graph on Poisson points on a finite collection of disjoint intervals.

math.PR↗

arXiv2007

Logarithmic speeds for one-dimensional perturbed random walk in random environment

We study the random walk in random environment on {0,1,2,...}, where the environment is subject to a vanishing (random) perturbation. The two particular cases we consider are: (i) random walk in random environment perturbed from Sinai's regime; (ii) simple random walk with random perturbation. We give almost sure results on how far the random walker will be from the origin after a long time t, for almost every environment. We give both upper and lower almost sure bounds. These bounds are of order $(\log t)^β$, for $β\in (1,\infty)$, depending on the perturbation. In addition, in the ergodic cases, we give results on the rate of decay of the stationary distribution.

math.PR↗

arXiv2007

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

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 $\R^d$, $d\in\N$. 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.

math.PR↗

arXiv2006

Random walk in random environment with asymptotically zero perturbation

We give criteria for ergodicity, transience and null recurrence for the random walk in random environment on {0,1,2,...}, with reflection at the origin, where the random environment is subject to a vanishing perturbation. Our results complement existing criteria for random walks in random environments and for Markov chains with asymptotically zero drift, and are significantly different to these previously studied cases. Our method is based on a martingale technique - the method of Lyapunov functions.

math.PR↗

arXiv2006

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 R^d 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 α>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.

math.PR↗

arXiv2004

On the total length of the random minimal directed spanning tree

In Bhatt and Roy's minimal directed spanning tree (MDST) construction for a random partially ordered set of points in the unit square,all edges must respect the ``coordinatewise'' partial order and there must be a directed path from each vertex to a minimal element. We study the asymptotic behaviour of the total length of this graph with power weighted edges. The limiting distribution is given by the sum of a normal component away from the boundary and a contribution introduced by the boundary effects, which can be characterized by a fixed point equation, and is reminiscent of limits arising in the probabilistic analysis of certain algorithms. As the exponent of the power weighting increases, the distribution undergoes a phase transition from the normal contribution being dominant to the boundary effects dominating. In the critical case where the weight is simple Euclidean length, both effects contribute significantly to the limit law. We also give a law of large numbers for the total weight of the graph.

math.PR↗