Asymptotic Arguments Research Articles

Multicasting in Stochastic MIMO Networks

Physical-layer multicast transmission, which seeks to deliver information messages to all users simultaneously, is becoming more important in wireless systems with demand for various multimedia mobile applications such as Multimedia Broadcast Multicast Services. The spatial randomness of communicating nodes in a wireless network is one of inevitable uncertainties in the design and analysis of network information flow and connectivity. The use of multiple antennas at both transmitting and receiving nodes is the most promising strategy to increase spectral efficiency and communication reliability as well as to enhance physical-layer confidentiality of wireless systems. In this paper, we characterize multicasting in such a stochastic multiple-input multiple-output (MIMO) network where a probe transmitter broadcasts confidential data with sectorized transmission to legitimate receivers sitting in a region R. We first put forth a measure of the total amount of information flow, called the space—time capacity, into R in a spatial random field of legitimate receivers without accounting for intrinsic confidentiality at the physical layer. We then derive the space—time capacity into the sectoral region R and the nth nearest ergodic capacity in a Poisson field to characterize the spatial average and ordering of MIMO ergodic capacity achieved by legitimate receivers in R. Using the Mar\u{c}enko—Pastur law, we further assess the asymptotic space—time capacity and the nth nearest ergodic capacity per receive antenna as the antenna numbers tend to infinity. In the presence of eavesdropping, we determine a total amount of confidential information flow per receive antenna, called the space—time secrecy rate, into R in Poisson fields of receiving equivalents—with asymptotic arguments. Using an asymptotic secrecy graph on R, we also characterize local confidential connectivity such as the secrecy range, out-degree, and out-isolation probability of the probe transmitter. The framework developed in this work enables us to quantify the local information flow in random MIMO wireless networks by averaging first small-scale fading processes over time and then large-scale path losses over space.

Geometric covering arguments and ergodic theorems for free groups

We present a new approach to the proof of ergodic theorems for actions of non-amenable groups, and give here a complete self-contained account of it in the case of free groups. Our approach is based on direct geometric covering arguments and asymptotic invariance arguments generalizing those developed in the ergodic theory of amenable groups. The results we describe go beyond those previously established for measure-preserving actions of free groups, and demonstrate the significant role the boundary action of the free group plays in the ergodic theory of its measure-preserving actions. Furthermore, our approach suggests the possibility of putting the ergodic theory of amenable groups and nonamenable groups on an equal footing : both can be viewed as special cases in the general ergodic theory of amenable ergodic equivalence relations with finite invariant measure.

Slipping motion of large neutrally buoyant particles in turbulence

AbstractDirect numerical simulations are used to investigate the individual dynamics of large spherical particles suspended in a developed homogeneous turbulent flow. A definition of the direction of the particle motion relative to the surrounding flow is introduced and used to construct the mean fluid velocity profile around the particle. This leads to an estimate of the particle slipping velocity and its associated Reynolds number. The flow modifications due to the particle are then studied. The particle is responsible for a shadowing effect that occurs in the wake up to distances of the order of its diameter: the particle calms turbulent fluctuations and reduces the energy dissipation rate compared to its average value in the bulk. Dimensional arguments are presented to draw an analogy between particle effects on turbulence and wall flows. Evidence is obtained for the presence of a logarithmic sublayer at distances between the thickness of the viscous boundary layer and the particle diameter ${D}_{p} $. Finally, asymptotic arguments are used to relate the viscous sublayer quantities to the particle size and the properties of the outer turbulence. It is shown in particular that the skin-friction Reynolds number behaves as $R{e}_{\tau } \propto {({D}_{p} / \eta )}^{4/ 3} $.

(In-)stability of singular equivariant solutions to the Landau–Lifshitz–Gilbert equation

In this paper, we use formal asymptotic arguments to understand the stability properties of equivariant solutions to the Landau–Lifshitz–Gilbert model for ferromagnets. We also analyse both the harmonic map heatflow and Schrödinger map flow limit cases. All asymptotic results are verified by detailed numerical experiments, as well as a robust topological argument. The key result of this paper is that blowup solutions to these problems are co-dimension one and hence both unstable and non-generic.

Analytical Results for Front Pinning between an Hexagonal Pattern and a Uniform State in Pattern-Formation Systems

In pattern-forming systems, localized patterns are states of intermediate complexity between fully extended ordered patterns and completely irregular patterns. They are formed by stationary fronts enclosing an ordered pattern inside an homogeneous background. In two dimensions, the ordered pattern is most often hexagonal and the conditions for fronts to stabilize are still unknown. In this Letter, we show how the locking of these fronts depends on their orientation relative to the pattern. The theory rests on general asymptotic arguments valid when the spatial scale of the front is slow compared to that of the hexagonal pattern. Our analytical results are confirmed by numerical simulations with the Swift-Hohenberg equation, relevant to hydrodynamical and buckling instabilities, and a nonlinear optical cavity model.

A note on approximate Bayesian credible sets based on modified loglikelihood ratios

Higher-order asymptotic arguments for a scalar parameter of interest have been widely investigated for Bayesian inference. In this paper the theory of asymptotic expansions is discussed for a vector parameter of interest. A modified loglikelihood ratio is suggested, which can be used to derive approximate Bayesian credible sets with accurate frequentist coverage. Three examples are illustrated.

Disentangling Continuous Volatility From Jumps in Long-Run Risk-Return Relationships

Realized variance can be broken down into continuous volatility and jumps. We show that these two components have very different predictive powers on future long-term excess stock market returns. While continuous volatility is a key driver of medium to long-term risk-return relationships, jumps do not predict future medium to long-term excess returns. We use inference methods robust to persistent predictors in a multi-horizon setup. That is, we use a rescaled Student-t to test for significant risk-return links, give asymptotic arguments, and simulate its exact behavior under the null in the case of multiple regressors with different degrees of persistence. Then, with Wald tests of equality of the risk-return relationship at multiple horizons, we find no evidence against a proportional relationship, constant across horizons, between long-term continuous volatility and future returns. Two by-products of our analysis are that imposing model-based constraints on long term regressions can improve their efficiency, and short-run estimates are sensitive to short-term variability of the predictors.

Robust testing for stationarity of global surface temperature

Surface temperature is a major indicator of climate change. To test for the presence of an upward trend in surface-temperature (global warming), sophisticated statistical methods are typically used which depend on implausible and/or unverifiable assumptions, in particular on the availability of a very large number of measurements. In this paper, the validity of these methods is challenged. It is argued that the available series are simply not long enough to justify the use of methods which are based on asymptotic arguments, because only a small fraction of the information contained in the data is utilizable to distinguish between a trend and natural variability. Thus, a simple frequency-domain test is proposed for the case when all but a very small number of frequencies may be corrupted by transitory fluctuations. Simulations confirm its robustness against short-term autocorrelation. When applied to a global surface-temperature series, significance can be achieved with far fewer frequencies than required by conventional tests.

Design and Analysis of Compressed Sensing Radar Detectors

We consider the problem of target detection from a set of Compressed Sensing (CS) radar measurements corrupted by additive white Gaussian noise. We propose two novel architectures and compare their performance by means of Receiver Operating Characteristic (ROC) curves. Using asymptotic arguments and the Complex Approximate Message Passing (CAMP) algorithm, we characterize the statistics of the ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm reconstruction error and derive closed form expressions for both the detection and false alarm probabilities of both schemes. Of the two architectures, we demonstrate that the best performing one consists of a reconstruction stage based on CAMP followed by a detector. This architecture, which outperforms the ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -based detector in the ideal case of known background noise, can also be made fully adaptive by combining it with a conventional Constant False Alarm Rate (CFAR) processor. Using the state evolution framework of CAMP, we also derive Signal to Noise Ratio (SNR) maps that, together with the ROC curves, can be used to design a CS-based CFAR radar detector. Our theoretical findings are confirmed by means of both Monte Carlo simulations and experimental results.

A Generalization of the Aumann-Shapley Value for Risk Capital Allocation Problems

This paper analyzes risk capital allocation problems. For risk capital allocation problems, the aim is to allocate the risk capital of a firm to its divisions. Risk capital allocation is of central importance in risk-based performance measurement. We consider a case in which the aggregate risk capital is determined via a coherent risk measure. The academic literature advocates an allocation rule that, in game-theoretic terms, is equivalent to using the Aumann-Shapley value as solution concept. This value is however not well-defined in case a differentiability condition is not satisfied. As an alternative, we introduce an allocation rule inspired by the Shapley value in a fuzzy setting. We take a grid on a fuzzy participation set, define paths on this grid and construct an allocation rule based on a path. Then, we define a rule as the limit of the average over these allocations, when the grid size converges to zero. We introduce this rule for a broad class of coherent risk measures. We show that if the Aumann-Shapley value is well-defined, the allocation rule coincides with it. If the Aumann-Shapley value is not defined, which is due to non-differentiability problems, the allocation rule specifies an explicit allocation. It corresponds with the Mertens value, which is originally characterized in an axiomatic way (Mertens, 1988), whereas we provide an asymptotic argument.

Modelos não lineares para a liberação de potássio de estercos animais em latossolos

Modelos não lineares são adequados para a descrição da liberação de nutrientes, uma vez que estimam quantidades de interesse prático e apresentam boa qualidade de ajuste. Embora seu processo inferencial seja baseado em argumentos assintóticos, existem meios de se conhecer a intensidade da não linearidade. Neste trabalho, avaliou-se a não linearidade de dois modelos de regressão não linear por meio das curvaturas de Bates e Watts, vício de Box e do estudo das propriedades amostrais dos estimadores de mínimos quadrados, obtido por simulação. Os dados são provenientes do estudo, ao longo do tempo, da liberação de K de quatro estercos animais em combinação com dois solos. O modelo Exponencial foi mais adequado, em termos inferenciais e para aplicação prática, uma vez que por todas as medidas apresentou menor não linearidade.

Resolving a paradox of anomalous scalings in the diffusion of granular materials

Granular materials do not perform Brownian motion, yet diffusion can be observed in such systems when agitation causes inelastic collisions between particles. It has been suggested that axial diffusion of granular matter in a rotating drum might be "anomalous" in the sense that the mean squared displacement of particles follows a power law in time with exponent less than unity. Further numerical and experimental studies have been unable to definitively confirm or disprove this observation. We show two possible resolutions to this apparent paradox without the need to appeal to anomalous diffusion. First, we consider the evolution of arbitrary (non-point-source) initial data towards the self-similar intermediate asymptotics of diffusion by deriving an analytical expression for the instantaneous collapse exponent of the macroscopic concentration profiles. Second, we account for the concentration-dependent diffusivity in bidisperse mixtures, and we give an asymptotic argument for the self-similar behavior of such a diffusion process, for which an exact self-similar analytical solution does not exist. The theoretical arguments are verified through numerical simulations of the governing partial differential equations, showing that concentration-dependent diffusivity leads to two intermediate asymptotic regimes: one with an anomalous scaling that matches the experimental observations for naturally polydisperse granular materials, and another with a "normal" diffusive scaling (consistent with a "normal" random walk) at even longer times.

Robust Deviance Information Criterion for Latent Variable Models

It is shown in this paper that the data augmentation technique undermines the theoretical underpinnings of the deviance information criterion (DIC), a widely used information criterion for Bayesian model comparison, although it facilitates parameter estimation for latent variable models via Markov chain Monte Carlo (MCMC) simulation. Data augmentation makes the likelihood function non-regular and hence invalidates the standard asymptotic arguments. A robust form of DIC, denoted as RDIC, is advocated for Bayesian comparison of latent variable models. RDIC is shown to be a good approximation to DIC without data augmentation. While the later quantity is dicult to compute, the expectation { maximization (EM) algorithm facilitates the computation of RDIC when the MCMC output is available. Moreover, RDIC is robust to nonlinear transformations of latent variables and distributional representations of model specication. The proposed approach is applied to several popular models in economics and

Small sample uncertainty aspects in relation to bullwhip effect measurement

The bullwhip effect as a concept has been known for almost half a century starting with the Forrester effect. The bullwhip effect is “allegedly” observed in many supply chains, and it is generally accepted as a potential malice. Despite this the bullwhip effect still seems to be first and foremost a conceptual phenomenon. Some even simply denies that the phenomenon exists at all in practice. This of course makes it important to perform measurements because only “what gets measured gets done” as the saying goes in analytics. Few measurements are reported in the literature, however, typically based on standard statistical assumptions. In the case of a standard measurement of the bullwhip effect independence amongst the participating variables are required amongst others, but this is definitely known not to be perfectly true if any systematic control at all have taken place in the supply chain. This paper analyses the bullwhip measurement implications in case the standard test assumptions are violated and illustrates how to improve on the testing setup. This is further done with a special emphasis on the unavoidable small-sample aspects relating to such measurement in practice, which typically renders all statistical asymptotic or robust arguments quite unusable. It is shown how H0-95%-confidence test intervals still easily can be obtained numerically in such cases.

Non‐parametric smoothing and prediction for nonlinear circular time series

Not much research has been done in the field of circular time‐series analysis. We propose a non‐parametric theory for smoothing and prediction in the time domain for circular time‐series data. Our model is based on local constant and local linear fitting estimates of a minimizer of an angular risk function. Both asymptotic arguments and empirical examples are used to describe the accuracy of our methods.

Blinded Continuous Monitoring of Nuisance Parameters in Clinical Trials

SummaryDetermination of a clinical trial's size is an important task in the planning of any trial because of the direct implications of the sample size on feasibility, costs and timelines. However, sample size calculations are often subject to substantial uncertainty due to limited prior information on the size of nuisance parameters such as variances or event rates. Continuous monitoring of the nuisance parameter in clinical trials has been proposed as a tool to size trials appropriately. With this approach, the nuisance parameter is continuously monitored during the trial. The trial is stopped when the actual estimate for the nuisance parameter and sample size fulfil a stopping criterion. Continuous monitoring can therefore be viewed as a stochastic process with stopping time. We describe the bias that occurs with unblinded continuous monitoring of the variance in clinical trials by means of a simulation study. Then we propose a procedure for blinded continuous monitoring that does not require breaking the treatment code during the on-going study and show that the procedure does not suffer from the same biases as observed in unblinded monitoring. Results on the performance properties of such designs are given and the designs are compared with blinded re-estimation procedures with a single data look. By means of asymptotic theoretical arguments and finite sample size simulations we find that the variability in sample size is smaller with blinded continuous monitoring than with blinded sample size re-estimation whenever the power for both designs is close to the target value. Repeated sample size re-estimation is in between continuous monitoring and sample size re-estimation in this respect. Furthermore, we present a hypertension trial where blinded sample size re-estimation with a single data look was applied and we investigate the properties of blinded continuous monitoring in this setting. Finally we close with a brief discussion.

Shapes of free resolutions over a local ring

We classify the possible shapes of minimal free resolutions over a regular local ring. This illustrates the existence of free resolutions whose Betti numbers behave in surprisingly pathological ways. We also give an asymptotic characterization of the possible shapes of minimal free resolutions over hypersurface rings. Our key new technique uses asymptotic arguments to study formal Q-Betti sequences.

Control Variates for Estimation Based on Reversible Markov Chain Monte Carlo Samplers

Summary A general methodology is introduced for the construction and effective application of control variates to estimation problems involving data from reversible Markov chain Monte Carlo samplers. We propose the use of a specific class of functions as control variates, and we introduce a new consistent estimator for the values of the coefficients of the optimal linear combination of these functions. For a specific Markov chain Monte Carlo scenario, the form and proposed construction of the control variates is shown to provide an exact solution of the associated Poisson equation. This implies that the estimation variance in this case (in the central limit theorem regime) is exactly zero. The new estimator is derived from a novel, finite dimensional, explicit representation for the optimal coefficients. The resulting variance reduction methodology is primarily (though certainly not exclusively) applicable when the simulated data are generated by a random-scan Gibbs sampler. Markov chain Monte Carlo examples of Bayesian inference problems demonstrate that the corresponding reduction in the estimation variance is significant, and that in some cases it can be quite dramatic. Extensions of this methodology are discussed and simulation examples are presented illustrating the utility of the methods proposed. All methodological and asymptotic arguments are rigorously justified under essentially minimal conditions.

HYDRO-ELASTIC WAVES IN A COCHLEAR MODEL: NUMERICAL SIMULATIONS AND AN ANALYTICALLY REDUCED MODEL

A three-dimensional model of the hydro-elastic waves in the mammalian cochlea is presented along with numerical simulations. The cochlear fluid is treated as linear, incompressible, and inviscid. The cochlear partition is treated as a massless thin plate loaded by the fluid. This model is then reformulated by analytically removing the fluid variable with the use of a Dirichlet-to-Neumann operator. The resulting fifth-order nonlocal PDE for the motion of the partition is simulated using a novel implicit numerical scheme. Simulations demonstrate that this model exhibits traveling wave characteristics and a clear place principle. Asymptotic analysis in the small aspect ratio of the cochlea is performed on the given model equations with energetic concerns in mind. The results of simulations along with these asymptotic arguments suggest a relationship between the form and function of the cochlea which we compare to physiological data.

Spurious regression and lurking variables

We present asymptotic and finite-sample arguments to study the spurious regression problem. This problem may be solved by introducing a lurking variable in the specification even if it is merely a proxy variable. Moreover, this approach is also valid if the lurking variable is a trending mechanism, as when the spurious regression is due to nonstationarities in the variables.