Nonlinear weighted best simultaneous approximation in Banach spaces

The present paper is concerned with the problem of weighted best simultaneous approximations in Banach spaces. The weighted best simultaneous approximations to sequences from S- and BS-suns in the Banach space are characterized in view of the Kolmogorov conditions. Applications are provided for weighted best simultaneous approximations from RS-sets and strict RS-sets. Our results obtained in the present paper extend and improve all earlier known results in this direction.     

线性过程的强逼近

该文对由独立同分布随机变量序列所生成的线性过程建立了泛函重对数律和用Wiener过程对线性过程的强逼近结果.     

Approximations for a multivariate hybrid process with applications to change-point detection

We obtain probability inequalities and almost sure rates for the approximations of the hybrids of empirical and partial sums processes in the multivariate case. Applications to weighted bootstrap empirical processes as well as to change-point detection tests for general nonparametric regression models are discussed.     

Moment Solutions for the State Exiting Counting Processes of a Markov Renewal Process

Important performance measures for many Markov renewal processes are the counts of the exits from each state. We present solutions for the conditional first, second, and covariance moments of the state exiting counting processes for a Markov renewal process, and solutions for the unconditional equilibrium versions of the moments. We demonstrate the relationship between the conditional first moments for the state exiting and the state entering counting processes. For analytical and illustrative purposes, we concentrate on the two state case. Two asymptotic expansions for the moment functions are proposed and evaluated both analytically and empirically. The two approximations are shown to be competitive in terms of absolute relative error, but the second approximation has a simpler analytical form which is useful in analyzing more complex stochastic processes having an underlying MRP structure.     

Markov programming by successive approximations with respect to weighted supremum norms

Markovian decision processes are considered in the situation of discrete time, countable state space, and general decision space. By introducing a Banach space with a weighted supremum norm, conditions are derived, which guarantee convergence of successive approximations to the value function. These conditions are weaker then those required by the usual supnorm approach. Several properties of the successive approximations are derived.     

A study of renewal processes with infinite means

This study concerns the spent lifetime characteristic of renewal processes with infinite means. Recently, Mitov and Yanev established some important limit theorems on the asymptotic behavior of the spent lifetime which extend earlier classical results of Feller and Erickson. Here, we study the rates of convergence associated with these limit theorems by means of Monte Carlo simulation. We also identify the forms of finite approximations associated with the limits. Our simulation study leads to several questions of theoretical importance, which, if properly addressed, could open the way to applications in a variety of areas. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 3, pp. 386–402, July–September, 2006.     

Bayesian non-parametric estimation for age-dependent branching processes

The purpose of this paper is to obtain Bayes estimators for both the offspring and life-length distribution in the context of a Bellman-Harris age-dependent branching process. We take a non-parametric approach by letting the prior random distributions, for the offspring and life-length distributions, be independent Dirichlet processes. Our primary results concern the derivation of Bayes estimators, under weighted squared error loss for each distribution. We also indicate some of their asymptotic properties and briefly discuss the modifications that become necessary when the initial information is such that the prior random distribution cannot be taken to be independent.     

含估计参数的加权经验过程

在讨论(广义)非参数似然比拟合优度检验时,加权经验过程理论是一个非常重要的基础.但对含估计参数的加权经验过程理论,目前文献中很少讨论.对含估计参数的加权经验过程的上界型和积分型两种统计量的近似分布进行了讨论,给出了较一般的结果.所得结论叮以为进一步讨论复合零假设下(广义)非参似然比拟合优度检验提供理论基础.     

Rényi-type empirical processes

We develop a theory of asymptotics for Rényi-type weighted empirical and quantile processes and statistics via characterising their possible limiting behaviour in the middle and on the tails. In case of moderate weight functions tail limiting behaviour is found to be Gaussian, while heavily weighted tail empirical and uniform quantile processes are characterised by their respective Poisson process and exponential sums like asymptotic behaviour.     

Linear Programming Approximations for Markov Control Processes in Metric Spaces

We develop a general framework to analyze the convergence of linear-programming approximations for Markov control processes in metric spaces. The approximations are based on aggregation and relaxation of constraints, as well as inner approximations of the decision variables. In particular, conditions are given under which the control problems optimal value can be approximated by a sequence of finite-dimensional linear programs.     

A Radial Basis Function Scheme for Option Pricing in Exponential Lévy Models

Abstract

We use Radial Basis Function (RBF) interpolation to price options in exponential Lévy models by numerically solving the fundamental pricing PIDE (Partial integro-differential equations). Our RBF scheme can handle arbitrary singularities of the Lévy measure in 0 without introducing further approximations, making it simpler to implement than competing methods. In numerical experiments using processes from the CGMY-KoBoL class, the scheme is found to be second order convergent in the number of interpolation points, including for processes of unbounded variation.     

Approximations for Multi-Class Departure Processes

The exact analysis of a network of queues with multiple products is, in general, prohibited because of the non-renewal structure of the arrival and departure processes. Two-moment approximations (decomposition methods, Whitt [9] ) have been successfully used to study these systems. The performance of these methods, however, strongly depends on the quality of the approximations used to compute the squared coefficient variation (CV) of the different streams of products.In this paper, an approximation method for computing the squared coefficient of variation of the departure stream from a multi-class queueing system is presented. In particular, we generalize the results of Bitran and Tirupati [3] and Whitt [11] related to the interference effect.     

Asymptotics of Studentized <Emphasis Type="Italic">U</Emphasis>-type processes for changepoint problems

This paper investigates weighted approximations for Studentized U-statistics type processes, both with symmetric and antisymmetric kernels, only under the assumption that the distribution of the projection variate is in the domain of attraction of the normal law. The classical second moment condition E|h(X ₁, X ₂)|² < ∞ is also relaxed in both cases. The results can be used for testing the null assumption of having a random sample versus the alternative that there is a change in distribution in the sequence.     

Rough path limits of the Wong-Zakai type with a modified drift term

The Wong-Zakai theorem asserts that ODEs driven by “reasonable” (e.g. piecewise linear) approximations of Brownian motion converge to the corresponding Stratonovich stochastic differential equation. With the aid of rough path analysis, we study “non-reasonable” approximations and go beyond a well-known criterion of [Ikeda, Watanabe, North Holland, 1989] in the sense that our result applies to perturbations on all levels, exhibiting additional drift terms involving any iterated Lie brackets of the driving vector fields. In particular, this applies to the approximations by McShane ('72) and Sussmann ('91). Our approach is not restricted to Brownian driving signals. At last, these ideas can be used to prove optimality of certain rough path estimates.     

A diffusion process with killing: the time to formation of recurrent deleterious mutant genes

We analyze the role of recurrent mutation on the time to formation or detection of particular genotypes in finite populations. The traditional method of approximating Markov chain models by diffusion processes is used. However, the diffusion approximations that arise in this context are governed not only by infinitesimal drift and diffusion coefficients, but also by a state-dependent killing rate that arises from the formation events. The resulting processes are particularly tractable, and allow a comprehensive analysis of the role of mutation in this problem.     

Chernoff approximations of Feller semigroups in Riemannian manifolds

Chernoff approximations of Feller semigroups and the associated diffusion processes in Riemannian manifolds are studied. The manifolds are assumed to be of bounded geometry, thus including all compact manifolds and also a wide range of non-compact manifolds. Sufficient conditions are established for a class of second order elliptic operators to generate a Feller semigroup on a (generally non-compact) manifold of bounded geometry. A construction of Chernoff approximations is presented for these Feller semigroups in terms of shift operators. This provides approximations of solutions to initial value problems for parabolic equations with variable coefficients on the manifold. It also yields weak convergence of a sequence of random walks on the manifolds to the diffusion processes associated with the elliptic generator. For parallelizable manifolds this result is applied in particular to the representation of Brownian motion on the manifolds as limits of the corresponding random walks.     

Approximation of optimal stopping problems

We consider optimal stopping of independent sequences. Assuming that the corresponding imbedded planar point processes converge to a Poisson process we introduce some additional conditions which allow to approximate the optimal stopping problem of the discrete time sequence by the optimal stopping of the limiting Poisson process. The optimal stopping of the involved Poisson processes is reduced to a differential equation for the critical curve which can be solved in several examples. We apply this method to obtain approximations for the stopping of iid sequences in the domain of max-stable laws with observation costs and with discount factors.     

线性过程的强逼近和重对数律

本文讨论由独立同分布随机变量列产生的线性过程的泛函型重对数律和强逼近, 同时又给出由NA随机变量列产生的线性过程的重对数律.     

Computing Scan Statistic p Values Using Importance Sampling,With Applications to Genetics and Medical Image Analysis

We present an importance sampling method for deciding, based on an observed random field, if a scan statistic provides significant evidence of increased activity in some localized region of time or space. Our method allows consideration of scan statistics based simultaneously on multiple scan geometries. Our approach yields an unbiased p value estimate whose variance is typically smaller than that of the naive hit-or-miss Monte Carlo technique when the p value is small. Furthermore, our p value estimate is often accurate for critical values that are not far enough in the tails of the null distribution to allow for accurate approximations via extreme value theory. The importance sampling approach unifies the analysis of various random field models, from (spatial) point processes to Gaussian random fields. For a scan statistic M, the method produces a p value of the form P[M ≥ τ] = Bρ, where B is the Bonferroni upper bound and the correction factor ρ measures the conservativeness of this upper bound. We present the application of our importance sampling estimator to multinomial sequences (molecular genetics), spatial point processes (digital mammography), and Gaussian random fields (PET scan brain imagery).     

Traces of harmonic functions,capacities, and traces of symmetric Markov processes

We derive explicit isomorphism formulas between weighted Dirichlet integrals for harmonic functions and boundary Dirichlet forms. Applications yield results on traces of Markov processes and convergence quasieverywhere of harmonic functions.