Dependence Between Extreme Values of Discrete and Continuous Time Locally Stationary Gaussian Processes

The maximum of a continuous, locally stationary Gaussian process which satisfies Bermans condition on the long range dependence is compared with the maximum of this process sampled at discrete time points. These two extreme values are asymptotically totally dependent if the grid of the discrete time points is sufficiently dense, and asymptotically independent if the the grid points are sparse.AMS 2000 Subject Classification. Primary—60F05, Secondary—60G15     

The dependence of extreme values of discrete and continuous time strongly dependent Gaussian processes

In this note, the asymptotic relation between the maximum of a continuous strongly dependent stationary Gaussian process and the maximum of this process sampled at discrete time points is studied. It is shown that these two extreme values are asymptotically totally dependent no matter what the grid of the discrete time points is.     

On Piterbarg Max-Discretisation Theorem for Standardised Maximum of Stationary Gaussian Processes

With motivation from Hüsler (Extremes 7:179–190, 2004) and Piterbarg (Extremes 7:161–177, 2004) in this paper we derive the joint limiting distribution of standardised maximum of a continuous, stationary Gaussian process and the standardised maximum of this process sampled at discrete time points. We prove that these two random sequences are asymptotically complete dependent if the grid of the discrete time points is sufficiently dense, and asymptotically independent if the grid is sufficiently sparse. We show that our results are relevant for computational problems related to discrete time approximation of the continuous time maximum.     

Maxima and minima of homogeneous Gaussian random fields over continuous time and uniform grids

ABSTRACT

In this paper, for centred homogeneous Gaussian random fields the joint limiting distributions of normalized maxima and minima over continuous time and uniform grids are investigated. It is shown that maxima and minima are asymptotic dependent for strongly dependent homogeneous Gaussian random field with the choice of sparse grid, Pickands' grid or dense grid, while for the weakly dependent Gaussian random field maxima and minima are asymptotically independent.     

Limit distribution of the sum and maximum from multivariate Gaussian sequences

In this paper we study the asymptotic joint behavior of the maximum and the partial sum of a multivariate Gaussian sequence. The multivariate maximum is defined to be the coordinatewise maximum. Results extend univariate results of McCormick and Qi. We show that, under regularity conditions, if the maximum has a limiting distribution it is asymptotically independent of the partial sum. We also prove that the maximum of a stationary sequence, when normalized in a special sense which includes subtracting the sample mean, is asymptotically independent of the partial sum (again, under regularity conditions). The limiting distributions are also obtained.     

Convergence of the variable two-step BDF time discretisation of nonlinear evolution problems governed by a monotone potential operator

The initial-value problem for a first-order evolution equation is discretised in time by means of the two-step backward differentiation formula (BDF) on a variable time grid. The evolution equation is governed by a monotone and coercive potential operator. On a suitable sequence of time grids, the piecewise constant interpolation and a piecewise linear prolongation of the time discrete solution are shown to converge towards the weak solution if the ratios of adjacent step sizes are close to 1 and do not vary too much.      

Discrete and Continuous Time Extremes of Gaussian Processes

Joint distribution of maximums of a Gaussian stationary process in continuous time and in uniform grid on the real axis is studied. When the grid is sufficiently sparse, maxima are asymptotically independent. When the grid is sufficiently tight, the maximums asymptotically coincide. In the boundary case which we call Pickands grid, the limit distribution is non-degenerate. It calculated in terms of a Pickands type constant.AMS 2000 Subject Classification. Primary—60G70, Secondary—60G15*Partially supported by the Scientific foundation of the Netherlands, RFFI grant 0401-00700 and grant DFG 436 RUS 113/722.     

Linear relations in time series models. I

A multiple time series is defined as the sum of an autoregressive process on a line and independent Gaussian white noise on a hyperplane that goes through the origin and intersects the line at a single point. This process is a multiple autoregressive time series in which the regression matrices satisfy suitable conditions. It is shown that the maximum likelihood estimates of the line and the autoregression coefficients can be obtained as the values that minimize a given function, and that the remaining maximum likelihood estimates can be computed as simple functions of the first ones. It is also shown that the maximum likelihood estimates are equivariant with respect to the group of bijective linear transformations.     

On the convergence on nonrectangular grids

The convergence properties of a full discrete approximations to the convection-diffusion equation is the subject of this paper. The full discrete scheme considered is of Lagrangian type: Euler Implicit on time and centered finite difference on space, and is defined using nonrectangular grids. We analyse this scheme under smoothness conditions on nonrectangular space-time grid. The main result establish the convergence of the approximations and we prove that the assumptions on the discrete spatial nodes movement are achieved if we consider the equidistribution principle.     

The Finite Difference Based Fast Adaptive Composite Grid Method

The fast adaptive composite grid (FAC) method is an iterative method for solving discrete boundary value problems on composite grids. McCormick introduced the method in [8] and considered the convergence behaviour for discrete problems resulting from finite volume element discretization on composite grids. In this paper we consider discrete problems resulting from finite difference discretization on composite grids. We distinguish between two obvious discretization approaches at the grid points on the interfaces between fine and coarse subgrids. The FAC method for solving such discrete problems is described. In the FAC method several intergrid transfer operators appear. We study how the convergence behaviour depends on these intergrid transfer operators. Based on theoretical insights, (quasi-)optimal intergrid transfer operators are derived. Numerical results illustrate the fast convergence of the FAC method using these intergrid transfer operators.     

On the Local Time of the Weighted Bootstrap and Compound Empirical Processes

This article is mainly concerned with the local times of the weighted bootstrap process. We prove a strong approximation theorem for the local time of the weighted bootstrap process by the local time of a Brownian bridge. We consider also the local time of the compound empirical processes that can be seen, asymptotically, as the local time of the convolution of two independent Gaussian processes.     

Infinitely Divisible Limit Processes for the Ewens Sampling Formula

The Ewens sampling formula in population genetics can be viewed as a probability measure on the group of permutations of a finite set of integers. Functional limit theory for processes defined through partial sums of dependent variables with respect to the Ewens sampling formula is developed. Using techniques from probabilistic number theory, it is shown that, under very general conditions, a partial sum process weakly converges in a function space if and only if the corresponding process defined through sums of independent random variables weakly converges. As a consequence of this result, necessary and sufficient conditions for weak convergence to a stable process are established. A counterexample showing that these conditions are not necessary for the one-dimensional convergence is presented. Very few results on the necessity part are known in the literature.     

Analysis of algebraic multigrid parameters for two-dimensional steady-state heat diffusion equations

In this work, it is provided a comparison for the algebraic multigrid (AMG) and the geometric multigrid (GMG) parameters, for Laplace and Poisson two-dimensional equations in square and triangular grids. The analyzed parameters are the number of: inner iterations in the solver, grids and unknowns. For the AMG, the effects of the grid reduction factor and the strong dependence factor in the coarse grid on the necessary CPU time are studied. For square grids the finite difference method is used, and for the triangular grids, the finite volume one. The results are obtained with the use of an adapted AMG1R6 code of Ruge and Stüben. For the AMG the following components are used: standard coarsening, standard interpolation, correction scheme (CS), lexicographic Gauss–Seidel and V-cycle. Comparative studies among the CPU time of the GMG, AMG and singlegrid are made. It was verified that: (1) the optimum inner iterations is independent of the multigrid, however it is dependent on the grid; (2) the optimum number of grids is the maximum number; (3) AMG was shown to be sensitive to both the variation of the grid reduction factor and the strong dependence factor in the coarse grid; (4) in square grids, the GMG CPU time is 20% of the AMG one.     

OU models based on positive and negative subordinate processes applying in SHIBOR time series analysis and derivative pricing – through discrete differential method

ABSTRACT

On account of that the OU models based on Gaussian process cannot describe the characteristics of peak, bias and asymmetric thick tail in SHIBOR time series, this paper replaces the Gaussian process in OU model with Levy process which can be decomposed into positive and negative subordinate processes, constructs OU model based on positive and negative subordinate processes. Methods parameter estimation and stochastic simulation were carried out by making discrete the stochastic differential equations into stochastic difference equations. The result shows that non-Gaussian OU process based on positive and negative subordinate processes not only fits the time series but also has better economic interpretation. The innovation of our research is to build a model of Non-Gaussian OU process based on positive and negative subordinate processes with less stochastic terms, and it provides an efficient tool for forecasting SHIBOR time series.     

Numerical analysis of a continuous Galerkin method for damped sine-Gordon equation

In this article, we discuss the numerical solution for the two-dimensional (2-D) damped sine-Gordon equation by using a space–time continuous Galerkin method. This method allows variable time steps and space mesh structures and its discrete scheme has good stability which are necessary for adaptive computations on unstructured grids. Meanwhile, it can easily get the higher-order accuracy in both space and time directions. The existence and uniqueness to the numerical solution are strictly proved and a priori error estimate in maximum-norm is given without any space–time grid conditions attached. Also, we prove that if the mesh in each time level is generated in a reasonable way, we can get the optimal order of convergence in both temporal and spatial variables. Finally, the convergence rates are presented and analyzed by some numerical experiments to illustrate the validity of the scheme.     

Testing Gaussian process with applications to super-resolution

This article introduces exact testing procedures on the mean of a Gaussian process X derived from the outcomes of ${\ell }_1$-minimization over the space of complex valued measures. The process X can be thought as the sum of two terms: first, the convolution between some kernel and a target atomic measure (mean of the process); second, a random perturbation by an additive centered Gaussian process. The first testing procedure considered is based on a dense sequence of grids on the index set of X and we establish that it converges (as the grid step tends to zero) to a randomized testing procedure: the decision of the test depends on the observation X and also on an independent random variable. The second testing procedure is based on the maxima and the Hessian of X in a grid-less manner. We show that both testing procedures can be performed when the variance is unknown (and the correlation function of X is known). These testing procedures can be used for the problem of deconvolution over the space of complex valued measures, and applications in frame of the Super-Resolution theory are presented. As a byproduct, numerical investigations may demonstrate that our grid-less method is more powerful (it detects sparse alternatives) than tests based on very thin grids.     

A central limit theorem for estimation in Gaussian stationary time series observed at unequally spaced times

The central limit theorem is proved for estimates of parameters which specify the covariance structure of a zero mean, stationary, Gaussian, discrete time series observed at unequally spaced times. The estimates considered are obtained by a single iteration from consistent estimates. The result also applies to the maximum likelihood estimate if it is consistent although consistency is not proved here. The essential condition on the sampling times is that the finite sample information matrix, when divided by the sample size, has a limit which is nonsingular and has finite norm. Some examples are presented to illustrate this condition.     

Iterative procedures for solutions of random operator equations in Banach spaces

We construct random iterative processes for weakly contractive and asymptotically nonexpansive random operators and study necessary conditions for the convergence of these processes. It is shown that they converge to the random fixed points of these operators in the setting of Banach spaces. We also proved that an implicit random iterative process converges to the common random fixed point of a finite family of asymptotically quasi-nonexpansive random operators in uniformly convex Banach spaces.     

On the complexity and approximability of some Euclidean optimal summing problems

The complexity status of several well-known discrete optimization problems with the direction of optimization switching from maximum to minimum is analyzed. The task is to find a subset of a finite set of Euclidean points (vectors). In these problems, the objective functions depend either only on the norm of the sum of the elements from the subset or on this norm and the cardinality of the subset. It is proved that, if the dimension of the space is a part of the input, then all these problems are strongly NP-hard. Additionally, it is shown that, if the space dimension is fixed, then all the problems are NP-hard even for dimension 2 (on a plane) and there are no approximation algorithms with a guaranteed accuracy bound for them unless P = NP. It is shown that, if the coordinates of the input points are integer, then all the problems can be solved in pseudopolynomial time in the case of a fixed space dimension.     

Joint limit distributions of exceedances point processes and partial sums of gaussian vector sequence

In this paper, we study the joint limit distributions of point processes of exceedances and partial sums of multivariate Gaussian sequences and show that the point processes and partial sums are asymptotically independent under some mild conditions. As a result, for a sequence of standardized stationary Gaussian vectors, we obtain that the point process of exceedances formed by the sequence (centered at the sample mean) converges in distribution to a Poisson process and it is asymptotically independent of the partial sums. The asymptotic joint limit distributions of order statistics and partial sums are also investigated under different conditions.