Convergence in a multidimensional randomized Keynesian beauty contest

We study the asymptotics of a Markovian system of $N \geq 3$ particles in $[0,1]^d$ in which, at each step in discrete time, the particle farthest from the current centre of mass is removed and replaced by an independent $U [0,1]^d$ random particle. We show that the limiting configuration contains $N-1$ coincident particles at a random location $\xi_N \in [0,1]^d$. A key tool in the analysis is a Lyapunov function based on the squared radius of gyration (sum of squared distances) of the points. For d=1 we give additional results on the distribution of the limit $\xi_N$, showing, among other things, that it gives positive probability to any nonempty interval subset of $[0,1]$, and giving a reasonably explicit description in the smallest nontrivial case, N=3.Comment: 26 pages, 4 figure

Limiting behaviour of random spatial graphs and asymptotically homogeneous RWRE

We consider several random spatial graphs of the nearest-neighbour type, including the k- nearest neighbours graph, the on-line nearest-neighbour graph, and the minimal directed spanning tree. We study the large sample asymptotic behaviour of the total length of these graphs, with power-weighted edges. We give laws of large numbers and weak convergence results. We evaluate limiting constants explicitly. In Bhatt and Roy's minimal directed spanning tree (MDST) construction on random points in (0,1)(^2)， each point is joined to its nearest neighbour in the south-westerly direction. We show that the limiting total length (with power-weighted egdes) of the edges joined to the origin converges in distribution to a Dickman-type random variable. We also study the length of the longest edge in the MDST. For the total weight of the MDST, we give a weak convergence result. The limiting distribution is given a normal component plus a contribution due to boundary effects, which can be characterized by a fixed point equation. There is a phase transition in the limit law as the weight exponent increases. In the second part of this thesis, we give criteria for ergodicity, transience and null recurrence for the random walk in random environment (RWRE) on z+ = {0,1,2,...}, with reflection at the origin, where the random environment is subject to a vanishing perturbation from the so-called Sinai's regime. 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 一 the method of Lyapunov functions

Rate of escape and central limit theorem for the supercritical Lamperti problem

The study of discrete-time stochastic processes on the half-line with mean drift at x given by μ1(x)→0 as x→∞ is known as Lamperti’s problem. We give sharp almost-sure bounds for processes of this type in the case where μ1(x) is of order x−β for some β∈(0,1). The bounds are of order t1/(1+β), so the process is super-diffusive but sub-ballistic (has zero speed). We make minimal assumptions on the moments of the increments of the process (finiteness of (2+2β+ε)-moments for our main results, so fourth moments certainly suffice) and do not assume that the process is time-homogeneous or Markovian. In the case where xβμ1(x) has a finite positive limit, our results imply a strong law of large numbers, which strengthens and generalizes earlier results of Lamperti and Voit. We prove an accompanying central limit theorem, which appears to be new even in the case of a nearest-neighbour random walk, although our result is considerably more general. This answers a question of Lamperti. We also prove transience of the process under weaker conditions than those that we have previously seen in the literature. Most of our results also cover the case where β=0. We illustrate our results with applications to birth-and-death chains and to multi-dimensional non-homogeneous random walks

Logarithmic speeds for one-dimensional perturbed random walks in random environments

We study the random walk in a random environment on Z+={0,1,2,…}, where the environment is subject to a vanishing (random) perturbation. The two particular cases that we consider are: (i) a random walk in a random environment perturbed from Sinai’s regime; (ii) a simple random walk with a random perturbation. We give almost sure results on how far the random walker is from the origin, for almost every environment. We give both upper and lower almost sure bounds. These bounds are of order (logt)β, for β∈(1,∞), depending on the perturbation. In addition, in the ergodic cases, we give results on the rate of decay of the stationary distribution

Deposition, diffusion, and nucleation on an interval

Motivated by nanoscale growth of ultra-thin films, we study a model of deposition, on an interval substrate, of particles that perform Brownian motions until any two meet, when they nucleate to form a static island, which acts as an absorbing barrier to subsequent particles. This is a continuum version of a lattice model studied in the applied literature. We show that the associated interval-splitting process converges in the sparse deposition limit to a Markovian process (in the vein of Brennan and Durrett) governed by a splitting density with a compact Fourier series expansion but, apparently, no simple closed form. We show that the same splitting density governs the fixed deposition rate, large time asymptotics of the normalized gap distribution, so these asymptotics are independent of deposition rate. The splitting density is derived by solving an exit problem for planar Brownian motion from a right-angled triangle, extending work of Smith and Watson

Non-homogeneous random walks on a semi-infinite strip

We study the asymptotic behaviour of Markov chains (Xn,ηn) on Z+×S, where Z+ is the non-negative integers and S is a finite set. Neither coordinate is assumed to be Markov. We assume a moments bound on the jumps of Xn, and that, roughly speaking, ηn is close to being Markov when Xn is large. This departure from much of the literature, which assumes that ηn is itself a Markov chain, enables us to probe precisely the recurrence phase transitions by assuming asymptotically zero drift for Xn given ηn. We give a recurrence classification in terms of increment moment parameters for Xn and the stationary distribution for the large- X limit of ηn. In the null case we also provide a weak convergence result, which demonstrates a form of asymptotic independence between Xn (rescaled) and ηn. Our results can be seen as generalizations of Lamperti’s results for non-homogeneous random walks on Z+ (the case where S is a singleton). Motivation arises from modulated queues or processes with hidden variables where ηn tracks an internal state of the system

Limit theorems for random spatial drainage networks

Suppose that, under the action of gravity, liquid drains through the unit d-cube via a minimal-length network of channels constrained to pass through random sites and to flow with nonnegative component in one of the canonical orthogonal basis directions of Rd, d ≥ 2. The resulting network is a version of the so-called minimal directed spanning tree. We give laws of large numbers and convergence in distribution results on the large-sample asymptotic behaviour of the total power-weighted edge length of the network on uniform random points in (0, 1)d. The distributional results exhibit a weight-dependent phase transition between Gaussian and boundary-effect-derived distributions. These boundary contributions are characterized in terms of limits of the so-called on-line nearest-neighbour graph, a natural model of spatial network evolution, for which we also present some new results. Also, we give a convergence in distribution result for the length of the longest edge in the drainage network; when d = 2, the limit is expressed in terms of Dickman-type variables