The bivariate normal copula function is regularly varying

We derive the rate of decay of the tail dependence of the bivariate normal distribution and establish its link with regularly varying functions. This result is an initial step in explaining the discrepancy between a zero asymptotic tail dependence coefficient and mass in the tail of a joint distribution.     

Bridging between short-range and long-range dependence with mixed spatio-temporal Ornstein–Uhlenbeck processes

While short-range dependence is widely assumed in the literature for its simplicity, long-range dependence is a feature that has been observed in data from finance, hydrology, geophysics and economics. In this paper, we extend a Lévy-driven spatio-temporal Ornstein–Uhlenbeck process by randomly varying its rate parameter to model both short-range and long-range dependence. This particular set-up allows for non-separable spatio-temporal correlations which are desirable for real applications, as well as flexible spatial covariances which arise from the shapes of influence regions. Theoretical properties such as spatio-temporal stationarity and second-order moments are established. An isotropic g-class is also used to illustrate how the memory of the process is related to the probability distribution of the rate parameter. We develop a simulation algorithm for the compound Poisson case which can be used to approximate other Lévy bases. The generalized method of moments is used for inference and simulation experiments are conducted with a view towards asymptotic properties.     

Effects of orientation and interaction of grains on coercivity for sintered NdFeB magnets

The coercivity of NdFeB magnets is determined by the coercivity of individual grains and the interaction between the grains composed of the magnets. The coercivity of individual grains and the intergrain interaction depend on the degree of the grain alignment, “tanθ type” Gaussian function is applied to describing the degree of the grain alignment. According to different coercivity mechanisms, there are different formula on the coercivity and the angular dependence of coercivity. The interaction between grains can be classified as the long-range magnetostatic interaction and the exchange-coupling interaction of neighboring grains. For the sintered magnet, the grain size is large and the grain boundaries are mostly separated by the non-magnetic phase. So, the long-range magnetostatic interaction is much stronger than the exchange coupling interaction and it makes the coercivity of the magnet composed of misaligned grains be bigger than that of the magnet composed of ideally aligned grains. The effects of coercivity of individual grains and the intergrain interactions are taken into account, and the starting field theory is in agreement with the experimental result for the coercivity of sintered NdFeB magnets.     

GI/G/1排队系统的几种收敛速度

对随机模型,可以从不同角度研究其稳定性,一种是研究其转移概率函数趋向于平稳分布的速度,即各种遍历性;另一种是研究平稳分布的尾部衰减速度.本文从这两个方面着手,找它们之间的关系,对GI/G/1排队系统,给出等待时间列几何遍历、平稳分布轻尾与服务时间分布轻尾三者等价,l-遍历、平稳分布的尾部(l-1)-阶衰减与服务时间分布的尾部l-阶衰减三者等价,最后证明出等待时间列不是强遍历.     

Velocity fields with power-law spectra for modeling turbulent flows

We consider a generalization of homogeneous and isotropic Çinlar velocity fields to capture power-law spectra. The random velocity field is non-Gaussian with a representation motivated by Lagrangian and Eulerian observations. A wide range of turbulent flows can be generated by varying the stochastic parameters of the model. The velocity field being a functional version of Poisson shot-noise is constructed as the superposition of eddies randomized through their types and arrival times. We introduce a dependence between the eddy types which are spatial parameters and the decay parameter which is temporal. As a result, long-range correlation in space and a power-law spectrum previously used with Ornstein–Uhlenbeck velocity fields are achieved. We show that a corresponding power-law form for the probability distribution of the eddy diameter is sufficient for this result. The parameters of the probability distribution are further specified in view of Kolmogorov theory of the inertial scales. In particular, ∣k∣^−5/3 scaling of the spectrum is obtained. In the diffusive limit, we show that the parameters governing the decay and the arrival rate, and the speed of rotation of an eddy increase while its diameter decreases. That is, the eddies arrive fast, decay fast, and rotate fast with a small radius for a Brownian limit.     

Effects of orientation and interaction of grains on coercivity for sintered NdFeB magnets

The coercivity of NdFeB magnets is determined by the coercivity of individual grains and the interaction between the grains composed of the magnets. The coercivity of individual grains and the intergrain interaction depend on the degree of the grain alignment, “tanθ type” Gaussian function is applied to describing the degree of the grain alignment. According to different coercivity mechanisms, there are different formula on the coercivity and the angular dependence of coercivity. The interaction between grains can be classified as the long-range magnetostatic interaction and the exchange-coupling interaction of neighboring grains. For the sintered magnet, the grain size is large and the grain boundaries are mostly separated by the non-magnetic phase. So, the long-range magnetostatic interaction is much stronger than the exchange coupling interaction and it makes the coercivity of the magnet composed of misaligned grains be bigger than that of the magnet composed of ideally aligned grains. The effects of coercivity of individual grains and the intergrain interactions are taken into account, and the starting field theory is in agreement with the experimental result for the coercivity of sintered NdFeB magnets. Project supported by the National Natural Science Foundation of China (Grant No. 59571017).     

Characterization of the Reynolds-stress and dissipation-rate decay and anisotropy from DNS of grid-generated turbulence

Grid–generated turbulence is a classical but still controversial topic, one open issue being the spatial decay rate of turbulent energy. We study the influence of grid geometry on the Reynolds–stress and dissipation–rate tensors, including the range and exponent of their self–similar spatial decay. DNS using a validated lattice Boltzmann code at mean–flow Reynolds numbers up to 1400 are performed, comparing square grids with blockage ratios from 0.05 to 0.49. A clear picture of spatial distribution and self–similarity emerges for the statistics of interest: Axisymmetry is excellently confirmed. A consistent power law decay is found in the self–similar decay region beyond 10 grid stride lengths downstream. Its exponent of –5/3 can be obtained, for weak turbulence, from a spatial flux balance reminiscent of the constant transport through the inertial range of isotropic turbulence. In the near–grid region, on the other hand, differences in Reynolds stress components are pronounced while those between dissipation tensor components are only recognizable very close to the grid, where a strong dependence on grid porosity is found. A normalization with respect to porosity is proposed that collapses the data from all runs. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A continuous updating weighted least squares estimator of tail dependence in high dimensions

Likelihood-based procedures are a common way to estimate tail dependence parameters. They are not applicable, however, in non-differentiable models such as those arising from recent max-linear structural equation models. Moreover, they can be hard to compute in higher dimensions. An adaptive weighted least-squares procedure matching nonparametric estimates of the stable tail dependence function with the corresponding values of a parametrically specified proposal yields a novel minimum-distance estimator. The estimator is easy to calculate and applies to a wide range of sampling schemes and tail dependence models. In large samples, it is asymptotically normal with an explicit and estimable covariance matrix. The minimum distance obtained forms the basis of a goodness-of-fit statistic whose asymptotic distribution is chi-square. Extensive Monte Carlo simulations confirm the excellent finite-sample performance of the estimator and demonstrate that it is a strong competitor to currently available methods. The estimator is then applied to disentangle sources of tail dependence in European stock markets.     

A risky asset model based on Lévy processes and asymptotically self-similar activity time processes with long-range dependence

In the paper,using Lvy processes subordinated by‘asymptotically self-similar activity time’processes with long-range dependence,we set up new asset pricing models.Using the diferent construction for gamma(Γ)based‘asymptotically self-similar activity time’processes with long-range dependence from Finlay and Seneta(2006)we extend the constructions for inverse-gamma and gamma based‘asymptotically selfsimilar activity time’processes with integer-valued parameters and long-range dependence in Heyde and Leonenko(2005)and Finlay and Seneta(2006)to noninteger-valued parameters.     

Sample path moderate deviations for a family of long-range dependent traffic and associated queue length processes

We consider the long-range dependent cumulative traffic generated by the superposition of constant rate fluid sources having exponentially distributed intra start times and Pareto distributed durations with finite mean and infinite variance. We prove a sample path moderate deviation principle when the session intensity is increased and the processes are centered and scaled appropriately. The governing rate function is known from large deviation principles for the tail probabilities of fractional Brownian motion. We derive logarithmic tail asymptotics for associated queue length processes when the traffic loads an infinite buffer with constant service rate. The moderate deviation approximation of steady-state queue length tail probabilities is compared to those obtained by computer simulations.     

Large deviations in the reinforced random walk model on trees

In this paper, we consider the linearly reinforced and the once-reinforced random walk models in the transient phase on trees. We show the large deviations for the upper tails for both models. We also show the exponential decay for the lower tail in the once-reinforced random walk model. However, the lower tail is in polynomial decay for the linearly reinforced random walk model.     

Quantile Function Expansion Using Regularly Varying Functions

We present a simple result that allows us to evaluate the asymptotic order of the remainder of a partial asymptotic expansion of the quantile function h(u) as u → 0⁺ or 1^?. This is focussed on important univariate distributions when h(?) has no simple closed form, with a view to assessing asymptotic rate of decay to zero of tail dependence in the context of bivariate copulas. Motivation of this study is illustrated by the asymptotic behaviour of the tail dependence of Normal copula. The Normal, Skew-Normal and Gamma are used as initial examples. Finally, we discuss approximation to the lower quantile of the Variance-Gamma and Skew-Slash distributions.     

Models of stochastic geometry — A survey

This paper discusses some models of stochastic geometry which are of potential interest for operations research. These are the Boolean model, a certain model for random compact sets and marked point processes. The Boolean model is a generalization of the well-known queueing systemM/G/. The random compact set model may be useful for modelling spatial spreading processes such as fires, cancers or holes in the Earth's surface. Marked point processes are used here as models of forests and used for a statistical study of the spatial distribution of damaged trees.Extended version of an Invited Lecture on the 16th Symposium for OR in Hamburg 1992.     

Macroscopic models for long-range dependent network traffic

A common way to inject long-range dependence in a stochastic traffic model possessing a weak regenerative structure is to make the variance of the underlying period infinite (while keeping the mean finite). This method is supported both by physical reasoning and by experimental evidence. We exhibit the long-range dependence of such a process and, by studying its second-order properties, we asymptotically match its correlation structure to that of a fractional Brownian motion. By studying a certain distributional limit theorem associated with such a process, we explain the emergence of an extremely skewed stable Lévy motion as a macroscopic model for the aforementioned traffic. Surprisingly, long-range dependence vanishes in the limit, being “replaced” by independent increments and highly varying marginals. The marginal distribution is computed and is shown to match the one empirically obtained in practice. Results on performance of queueing systems with Lévy inputs of the aforementioned type are also reported in this paper: they are shown to be in agreement with pre-limiting models, without violating experimental queueing analysis. This revised version was published online in June 2006 with corrections to the Cover Date.     

Reinsurance control in a model with liabilities of the fractional Brownian motion type

We propose a model for reinsurance control for an insurance firm in the case where the liabilities are driven by fractional Brownian motion, a stochastic process exhibiting long-range dependence. The problem is transformed to a nonlinear programming problem, the solution of which provides the optimal reinsurance policy. The effect of various parameters of the model, such as the safety loading of the reinsurer and the insurer, the Hurst parameter, etc. on the optimal reinsurance program is studied in some detail. Copyright © 2007 John Wiley & Sons, Ltd.     

Spectral Properties of Uperpositions of Ornstein-Uhlenbeck Type Processes

Stationary processes with prescribed one-dimensional marginal laws and long-range dependence are constructed. The asymptotic properties of the spectral densities are studied. The possibility of Mittag-Leffler decay in the autocorrelation function of superpositions of Ornstein-Uhlenbeck type processes is proved. AMS 2000 Subject Classification Primary 60E07, 60G10, 60G18 Secondary 62M10, 62P05 Supported by the Danish National Research Foundation and EPSRC grant RCMT091. Partially supported by the ARC grants DP 0345577, DP0559807 and EPSRC grant RCMT091.     

The fractional multivariate normal tempered stable process

In this paper, the multivariate process having long-range dependency is presented. The process is defined by the time-changed fractional Brownian motion whose subordinator is given by the fractional tempered stable subordinator. The fractional tempered stable subordinator is a generalization of the non-decreasing tempered stable process with long-range dependence. The multivariate process allows for (1) the long-range dependence in the endogenous noise, (2) the long-range dependence in time or the volatility, (3) the fat-tailed marginal distribution, and (4) an asymmetric dependence structure between elements. Numerical methods to generating sample paths for the process are discussed.     

Minimax-rate adaptive nonparametric regression with unknown correlations of errors In Celebration of Professor Lincheng Zhao's 75th Birthday

Minimax-rate adaptive nonparametric regression has been intensively studied under the assumption of independent or uncorrelated errors in the literature. In many applications, however, the errors are dependent,including both short-and long-range dependent situations. In such a case, adaptation with respect to the unknown dependence is important. We present a general result in this direction under Gaussian errors. It is assumed that the covariance matrix of the errors is known to be in a list of specifications possibly including independence, short-range dependence and long-range dependence as well. The regression function is known to be in a countable(or uncountablu but well-structured) collection of function classes. Adaptive estimators are constructed to attain the minimax rate of convergence automatically for each function class under each correlation specification in the corresponding lists.     

Asymptotically (in)dependent multivariate maxima of moving maxima processes

Smith and Weissman introduced a M4 class of processes which are very flexible models for temporally dependent multivariate extreme value processes. However all variables in these M4 models are asymptotically dependent and what this paper does is to extend this M4 class in a number of ways to produce classes of models which are also asymptotically independent. We shall study properties of the proposed models. In particular, asymptotic dependence indexes, coefficients of tail dependence, and extremal indexes are derived for each case.     

Geometric tail of queue length of low-priority customers in a nonpreemptive priority MAP/PH/1 queue

We consider a MAP/PH/1 queue with two priority classes and nonpreemptive discipline, focusing on the asymptotic behavior of the tail probability of queue length of low-priority customers. A sufficient condition under which this tail probability decays asymptotically geometrically is derived. Numerical methods are presented to verify this sufficient condition and to compute the decay rate of the tail probability.