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    ABSTRACT: We study the asymptotic behaviour of Markov chains $(X_n,\eta_n)$ on $\mathbb{Z}_+ \times S$, where $\mathbb{Z}_+$ 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 $X_n$, and that, roughly speaking, $\eta_n$ is close to being Markov when $X_n$ is large. This departure from much of the literature, which assumes that $\eta_n$ is itself a Markov chain, enables us to probe precisely the recurrence phase transitions by assuming asymptotically zero drift for $X_n$ given $\eta_n$. We give a recurrence classification in terms of increment moment parameters for $X_n$ and the stationary distribution for the large-$X$ limit of $\eta_n$. In the null case we also provide a weak convergence result, which demonstrates a form of asymptotic independence between $X_n$ (rescaled) and $\eta_n$. Our results can be seen as generalizations of Lamperti's results for non-homogeneous random walks on $\mathbb{Z}_+$ (the case where $S$ is a singleton). Motivation arises from modulated queues or processes with hidden variables where $\eta_n$ tracks an internal state of the system.
    Stochastic Processes and their Applications. 02/2014;
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    ABSTRACT: The challenge of understanding complex systems often gives rise to a mul-tiplicity of models. It is natural to consider whether the outputs of these models can be combined to produce a system prediction that is more infor-mative than the output of any one of the models taken in isolation. And, in particular, to consider the relationship between the spread of model out-puts and system uncertainty. We describe a statistical framework for such a combination, based on the exchangeability of the models, and their co-exchangeability with the system. We demonstrate the simplest implemen-tation of our framework in the context of climate prediction. Throughout we work entirely in means and variances, to avoid the necessity of specifying higher-order quantities for which we often lack well-founded judgements.
    Journal of the American Statistical Association 09/2013; 108(503).
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    ABSTRACT: 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.
    Stochastic Processes and their Applications 04/2013;
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    ABSTRACT: The theory of classically integrable nonlinear wave equations and the Bethe ansatz systems describing massive quantum field theories defined on an infinite cylinder are related by an important mathematical correspondence that still lacks a satisfactory physical interpretation. In this paper, we shall extend this link to the case of the classical and quantum versions of the Tzitzéica-Bullough-Dodd model.
    Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences 01/2013; 371(1989):20120052.
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    ABSTRACT: We study discrete-time stochastic processes $(X_t)$ on $[0,\infty)$ with asymptotically zero mean drifts. Specifically, we consider the critical (Lamperti-type) situation in which the mean drift at $x$ is about $c/x$. Our focus is the recurrent case (when $c$ is not too large). We give sharp asymptotics for various functionals associated with the process and its excursions, including results on maxima and return times. These results include improvements on existing results in the literature in several respects, and also include new results on excursion sums and additive functionals of the form $\sum_{s \leq t} X_s^\alpha$, $\alpha >0$. We make minimal moments assumptions on the increments of the process. Recently there has been renewed interest in Lamperti-type process in the context of random polymers and interfaces, particularly nearest-neighbour random walks on the integers; some of our results are new even in that setting. We give applications of our results to processes on the whole of $\R$ and to a class of multidimensional `centrally biased' random walks on $\R^d$; we also apply our results to the simple harmonic urn, allowing us to sharpen existing results and to verify a conjecture of Crane et al.
    Stochastic Processes and their Applications. 08/2012; 123(6).
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    ABSTRACT: We show that polarons can sustain looplike configurations in flexible biopolymers and that the size of the loops depend on both the flexural rigidity of the polymer and the electron-phonon coupling constant. In particular we show that for single stranded DNA (ssDNA) and polyacetylene such loops can have as few as seven monomers. We also show that these configurations are very stable under thermal fluctuations and so could facilitate the formation of hairpin loops of ssDNA.
    Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics 04/2012; 86(2).
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    ABSTRACT: A species sensitivity distribution (SSD) models data on toxicity of a specific toxicant to species in a defined assemblage. SSDs are typically assumed to be parametric, despite noteworthy criticism, with a standard proposal being the log-normal distribution. Recently, and confusingly, there have emerged different statistical methods in the ecotoxicological risk assessment literature, independent of the distributional assumption, for fitting SSDs to toxicity data with the overall aim of estimating the concentration of the toxicant that is hazardous to % of the biological assemblage (usually with small). We analyze two such estimators derived from simple linear regression applied to the ordered log-transformed toxicity data values and probit transformed rank-based plotting positions. These are compared to the more intuitive and statistically defensible confidence limit-based estimator. We conclude based on a large-scale simulation study that the latter estimator should be used in typical assessments where a pointwise value of the hazardous concentration is required.
    Risk Analysis 11/2011; 32(7):1232-43.
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    ABSTRACT: We study kink-antikink collisions in the one-dimensional nonintegrable scalar φ⁶ model. Although the single-kink solutions for this model do not possess an internal vibrational mode, our simulations reveal a resonant scattering structure, thereby providing a counterexample to the standard belief that the existence of such a mode is a necessary condition for multibounce resonances in general kink-antikink collisions. We investigate the two-bounce windows in detail, and present evidence that this structure is caused by the existence of bound states in the spectrum of small oscillations about a combined kink-antikink configuration.
    Physical Review Letters 08/2011; 107(9):091602.
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    ABSTRACT: A pair of lower and upper cumulative distribution functions, also called probability box or p-box, is among the most popular models used in imprecise probability theory. They arise naturally in expert elicitation, for instance in cases where bounds are specified on the quantiles of a random variable, or when quantiles are specified only at a finite number of points. Many practical and formal results concerning p-boxes already exist in the literature. In this paper, we provide new efficient tools to construct multivariate p-boxes and develop algorithms to draw inferences from them. For this purpose, we formalise and extend the theory of p-boxes using Walley’s behavioural theory of imprecise probabilities, and heavily rely on its notion of natural extension and existing results about independence modeling. In particular, we allow p-boxes to be defined on arbitrary totally preordered spaces, hence thereby also admitting multivariate p-boxes via probability bounds over any collection of nested sets. We focus on the cases of independence (using the factorization property), and of unknown dependence (using the Fréchet bounds), and we show that our approach extends the probabilistic arithmetic of Williamson and Downs. Two design problems—a damped oscillator, and a river dike—demonstrate the practical feasibility of our results.
    International Journal of Approximate Reasoning 03/2011;
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    ABSTRACT: Transmission matrices for two types of integrable defect are calculated explicitly, first by solving directly the nonlinear transmission Yang–Baxter equations, and second by solving a linear intertwining relation between a finite-dimensional representation of the relevant Borel subalgebra of the quantum group underpinning the integrable quantum field theory and a particular infinite-dimensional representation expressed in terms of sets of generalised ‘quantum’ annihilation and creation operators. The principal examples analysed are based on the and affine Toda models but examples of similar infinite-dimensional representations for quantum Borel algebras for all other affine Toda theories are also provided.
    Nuclear Physics B 01/2011;
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