Joint asymptotic distribution of exceedances point process and partial sum of stationary Gaussian sequence

Let {X _i } _i=1^∞ be a standardized stationary Gaussian sequence with covariance function r(n) = EX ₁ X _n+1, S _n = Σ _i=1 ⁿ X _i , and $\bar X_n = \tfrac{{S_n }} {n} $\bar X_n = \tfrac{{S_n }} {n} . And let N _n be the point process formed by the exceedances of random level $(\tfrac{x} {{\sqrt {2\log n} }} + \sqrt {2\log n} - \tfrac{{\log (4\pi \log n)}} {{2\sqrt {2\log n} }})\sqrt {1 - r(n)} + \bar X_n $(\tfrac{x} {{\sqrt {2\log n} }} + \sqrt {2\log n} - \tfrac{{\log (4\pi \log n)}} {{2\sqrt {2\log n} }})\sqrt {1 - r(n)} + \bar X_n by X ₁,X ₂,…, X _n . Under some mild conditions, N _n and S _n are asymptotically independent, and N _n converges weakly to a Poisson process on (0,1].     

On the Bayesian estimation for the stationary Neyman-Scott point processes

The pure and modified Bayesian methods are applied to the estimation of parameters of the Neyman-Scott point process. Their performance is compared to the fast, simulation-free methods via extensive simulation study. Our modified Bayesian method is found to be on average 2.8 times more accurate than the fast methods in the relative mean square errors of the point estimates, where the average is taken over all studied cases. The pure Bayesian method is found to be approximately as good as the fast methods. These methods are computationally affordable today.     

On properties of central manifolds of a stationary point

We present results concerning properties of central manifolds of a stationary point. The results are illustrated by examples.     

On generalized stationary iterative method for solving the saddle point problems

In this paper, the generalized stationary iterative (GSI) method is studied for solving the saddle point problems. The convergence of this method is studied under suitable restrictions on iteration parameters. Numerical example for solving Stokes equation is presented to show the superiority of GSI method.     

On the Fourier series of a stationary process. II

Probability Theory and Related Fields - Let C n =C n (T) be the Fourier coefficients of a continuous weakly stationary process over (?1/2T,1/2T). We make some corrections of the results on...     

On the asymptotic location of high values of a stationary sequence

In this paper, we investigate the limiting distribution of the locations related with high values generated by a strictly stationary sequence of random variables. The main tool for this purpose is the so-called local extremes comparison lemma, which enables us to obtain the convergence in distribution of various functionals related with the location of extreme order statistics, including the location of local maxima and the joint locations of the largest order statistics. Furthermore, results about the joint asymptotic behavior of the location of the first high-level exceedance and the location of the maximum are also discussed.     

On selection of the order of the spectral density model for a stationary process

Summary Let {X(t)} be a stationary process with mean zero and spectral densityg(x). We shall use akth order parametric spectral modelf _τ(k) (x) for this process. Without Gaussianity we can obtain an estiamte of τ(k), say ĝt(k), by maximizing the quasi-Gaussian likelihood of this model. We can then construct the best linear predictor ofX(t), which is computed on the basis of the estimated spectral densityf _ĝt(k) (x). An asymptotic lower bound of the mean square error of the estimated predictor is obtained. The bound is attained ifk is selected by Akaike's information criterion.     

The empirical process of a short-range dependent stationary sequence under Gaussian subordination

Summary Consider the stationary sequenceX ₁=G(Z ₁),X ₂=G(Z ₂),..., whereG(·) is an arbitrary Borel function andZ ₁,Z ₂,... is a mean-zero stationary Gaussian sequence with covariance functionr(k)=E(Z ₁ Z _k+1) satisfyingr(0)=1 and _k=1 |r(k)| ^m < , where, withI{·} denoting the indicator function andF(·) the continuous marginal distribution function of the sequence {X _n }, the integerm is the Hermite rank of the family {I{G(·) x} –F(x):xR}. LetF _n (·) be the empirical distribution function ofX ₁,...,X _n . We prove that, asn, the empirical processn ^1/2{F _n (·)-F(·)} converges in distribution to a Gaussian process in the spaceD[–,].Partially supported by NSF Grant DMS-9208067     

On obtaining a stationary process isomorphic to a given process with a desired distribution

Let _e denote the set of distributions of all stationary, ergodic, aperiodic processes with a given finite state space, and let the metric on _e be Ornstein's process distance. Suppose is a subset of _e which is a in the weak topology and for which (µ _n ,)0 whenever { _n } is a sequence from _e converging weakly to a positive entropy measure in . It is shown that ifX is a stationary ergodic aperiodic process with entropy rate less than the entropy of one of the distributions in , thenX is isomorphic to a process whose distribution lies in . As special cases, one obtains the invulnerable source coding theorem of information theory and also the Grillenberger-Krengel theorem on the existence of a generator whose process has a desired marginal distribution.Research of author supported by NSF Grant ECS-78-21335.     

On the solution of a nonlinear stationary inhomogeneous filtration problem with a point source

We prove the convergence of the simple iteration method at the rate of a geometric progression in the norm of Sobolev and Hölder spaces to the solution of a stationary nonlinear inhomogeneous filtration problem with a point source.     

On the option valuation for a combined diffusion/point process model

This paper considers the price at time zero of a contingent claim when the price process is a diffusion/point process model. And we apply this price to European option.     

Finding a Nash equilibrium in noncooperativeN-person games by solving a sequence of linear stationary point problems

In this paper we present an algorithm for finding a Nash equilibrium in a noncooperative normal formN-person game. More generally, the algorithm can be applied for solving a nonlinear stationary point problem on a simplotope, being the Cartesian product of several simplices. The algorithm solves the problem by solving a sequence of linear stationary point problems. Each problem in the sequence is solved in a finite number of iterations. Although the overall convergence cannot be proved, the method performs rather well. Computational results suggest that this algorithm performs at least as good as simplicial algorithms do.For the special case of a bi-matrix game (N=2), the algorithm has an appealing game-theoretic interpretation. In that case, the problem is linear and the algorithm always finds a solution. Furthermore, the equilibrium found in a bi-matrix game is perfect whenever the algorithm starts from a strategy vector at which all actions are played with positive probability.This research is part of the VF-program Co-operation and Competition, which has been approved by the Netherlands Ministery of Education and Sciences.     

On estimating linear functional of the covariance function of a stationary process

A general estimator for linear functionals of the covariance function of a stationary process is investigated. The rate of convergence is obtained for a large class of functionals. The rate depends essentially on the functional. This yields an explanation of the different rates of convergence for various spectral estimates. The asymptotic normality of the estimator is proved under integrability conditions on the cumulant functions     

Almost sure limit theorems for a stationary normal sequence

We prove almost sure limit theorems for the maximum of a stationary normal sequence under some conditions.