Poisson process Fock space representation, chaos expansion and covariance inequalities

We consider a Poisson process ?? on an arbitrary measurable space with an arbitrary sigma-finite intensity measure. We establish an explicit Fock space representation of square integrable functions of ??. As a consequence we identify explicitly, in terms of iterated difference operators, the integrands in the Wiener?CIt? chaos expansion. We apply these results to extend well-known variance inequalities for homogeneous Poisson processes on the line to the general Poisson case. The Poincaré inequality is a special case. Further applications are covariance identities for Poisson processes on (strictly) ordered spaces and Harris?CFKG-inequalities for monotone functions of ??.     

Positive definite symmetric functions on finite-dimensional spaces II

Integral representations for the density functions of absolutely continuous α-symmetric random vectors are derived, and general methods for constructing new α-symmetric distributions are presented. An explicit formula, for determining the spectral measure of a symmetric stable random vector from its characteristic function, is obtained.     

Characterization of dependence of multidimensional Lévy processes using Lévy copulas

This paper suggests Lévy copulas in order to characterize the dependence among components of multidimensional Lévy processes. This concept parallels the notion of a copula on the level of Lévy measures. As for random vectors, a version of Sklar's theorem states that the law of a general multivariate Lévy process is obtained by combining arbitrary univariate Lévy processes with an arbitrary Lévy copula. We construct parametric families of Lévy copulas and prove a limit theorem, which indicates how to obtain the Lévy copula of a multivariate Lévy process X from the ordinary copula of the random vector X_t for small t.     

Vectors of two-parameter Poisson-Dirichlet processes

The definition of vectors of dependent random probability measures is a topic of interest in applications to Bayesian statistics. They represent dependent nonparametric prior distributions that are useful for modelling observables for which specific covariate values are known. In this paper we propose a vector of two-parameter Poisson-Dirichlet processes. It is well-known that each component can be obtained by resorting to a change of measure of a σ-stable process. Thus dependence is achieved by applying a Lévy copula to the marginal intensities. In a two-sample problem, we determine the corresponding partition probability function which turns out to be partially exchangeable. Moreover, we evaluate predictive and posterior distributions.     

Spectral Representation of Multivariate Regularly Varying Lévy and CARMA Processes

A spectral representation for regularly varying Lévy processes with index between one and two is established and the properties of the resulting random noise are discussed in detail, giving also new insight in the L ²-case where the noise is a random orthogonal measure. This allows a spectral definition of multivariate regularly varying Lévy-driven continuous time autoregressive moving average (CARMA) processes. It is shown that they extend the well-studied case with finite second moments and coincide with definitions previously used in the infinite variance case when they apply.     

Dilated Fractional Stable Motions

Dilated fractional stable motions are stable, self-similar, stationary increments random processes which are associated with dissipative flows. Self-similarity implies that their finite-dimensional distributions are invariant under scaling. In the Gaussian case, when the stability exponent equals 2, dilated fractional stable motions reduce to fractional Brownian motion. We suppose here that the stability exponent is less than 2. This implies that the dilated fractional stable motions have infinite variance and hence they cannot be characterised by a covariance function. These dilated fractional stable motions are defined through an integral representation involving a nonrandom kernel. This kernel plays a fundamental role. In this work, we study the space of kernels for which the dilated processes are well-defined, indicate connections to Sobolev spaces, discuss uniqueness questions and relate dilated fractional stable motions to other self-similar processes. We show that a number of processes that have been obtained in the literature, are in fact dilated fractional stable motions, for example, the telecom process obtained as limit of renewal reward processes, the Takenaka processes and the so-called random wavelet expansion processes.     

Properties of certain symmetric stable distributions

Necessary and sufficient conditions are presented for jointly symmetric stable random vectors to be independent and for a regression involving symmetric stable random variables to be linear. The notion of n-fold dependence is introduced for symmetric stable random variables, and under this condition we determine all monomials in such random variables for which moments exist.     

Local nondeterminism and local times for stable processes

Summary Our main theorem gives sufficient conditions for symmetric stable processes and fields to have a jointly continuous local time. The approach is through the L ^p representation for such processes. We develop a measure of dependence for vectors in a normed linear space and use that to analyze the probabilistic independence of the increments of a stable process. Local nondeterminism is defined for stable processes and shown to be equivalent to locally approximately independent increments. Sufficient conditions for several classes of stable processes to be local nondeterministic are given. These ideas are extended to multidimensional stable random fields and we prove existence of jointly continuous local times. The results extend most Gaussian results to their stable analogs.     

Extremes of Gaussian processes with a smooth random variance

Let ξ(t) be a standard stationary Gaussian process with covariance function r(t), and η(t), another smooth random process. We consider the probabilities of exceedances of ξ(t)η(t) above a high level u occurring in an interval [0,T] with T>0. We present asymptotically exact results for the probability of such events under certain smoothness conditions of this process ξ(t)η(t), which is called the random variance process. We derive also a large deviation result for a general class of conditional Gaussian processes X(t) given a random element Y.     

Stability analysis and stabilization of linear symmetric matrix-valued continuous,discrete, and impulsive dynamical systems — A unified approach for the stability analysis and the stabilization of linear systems

Matrix-valued dynamical systems are an important class of systems that can describe important processes such as covariance/second-order moment processes, or processes on manifolds and Lie Groups. We address here the case of processes that leave the cone of positive semidefinite matrices invariant, thereby including covariance and second-order moment processes. Both the continuous-time and the discrete-time cases are first considered. In the LTV case, the obtained stability and stabilization conditions are expressed as differential and difference Lyapunov conditions which are equivalent, in the LTI case, to some spectral conditions for the generators of the processes. Convex stabilization conditions are also obtained in both the continuous-time and the discrete-time setting. It is proven that systems with constant delays are stable provided that the systems with zero-delays are stable—which mirrors existing results for linear positive systems. The results are then extended and unified into an impulsive formulation for which similar results are obtained. The proposed framework is very general and can recover and/or extend many of the existing results in the literature on linear systems related to (mean-square) exponential (uniform) stability. Several examples are discussed to illustrate this claim by deriving stability conditions for stochastic systems driven by Brownian motion and Poissonian jumps, Markov jump systems, (stochastic) switched systems, (stochastic) impulsive systems, (stochastic) sampled-data systems, and all their possible combinations.     

Multivariate analysis of variance with fewer observations than the dimension

In this article, we consider the problem of testing a linear hypothesis in a multivariate linear regression model which includes the case of testing the equality of mean vectors of several multivariate normal populations with common covariance matrix Σ, the so-called multivariate analysis of variance or MANOVA problem. However, we have fewer observations than the dimension of the random vectors. Two tests are proposed and their asymptotic distributions under the hypothesis as well as under the alternatives are given under some mild conditions. A theoretical comparison of these powers is made.     

Martingale representation for Poisson processes with applications to minimal variance hedging

We consider a Poisson process η on a measurable space equipped with a strict partial ordering, assumed to be total almost everywhere with respect to the intensity measure λ of η. We give a Clark-Ocone type formula providing an explicit representation of square integrable martingales (defined with respect to the natural filtration associated with η), which was previously known only in the special case, when λ is the product of Lebesgue measure on R₊ and a σ-finite measure on another space X. Our proof is new and based on only a few basic properties of Poisson processes and stochastic integrals. We also consider the more general case of an independent random measure in the sense of Itô of pure jump type and show that the Clark-Ocone type representation leads to an explicit version of the Kunita-Watanabe decomposition of square integrable martingales. We also find the explicit minimal variance hedge in a quite general financial market driven by an independent random measure.     

Sample Covariance Matrix for Random Vectors with Heavy Tails

We compute the asymptotic distribution of the sample covariance matrix for independent and identically distributed random vectors with regularly varying tails. If the tails of the random vectors are sufficiently heavy so that the fourth moments do not exist, then the sample covariance matrix is asymptotically operator stable as a random element of the vector space of symmetric matrices.     

Existence of the linear prediction for Banach space valued Gaussian processes

The correspondence between Gaussian stochastic processes with values in a Banach space E and cylindrical processes which are related to them is studied. It is shown that the linear prediction of an E-valued Gaussian process is an E-valued random variable as well as the spectral measure of an E-valued Gaussian stationary process is a Gaussian random measure.     

A new method of spectral decomposition of covariance matrix in mixed effects models and its applications

For the mixed effects models with balanced data, a new ordering of design matrices of random effects is defined, and then a simple formula of the spectral decomposition of covariance matrix is obtained. To compare with the two methods in literature, the decomposition can not only give the actual number of all distinct eigenvalues and their expression, but also show clearly the relationship between the design matrices of random effects and the decomposition. These results can be applied to the problems for testifying the analysis of the variance estimate being a minimum variance unbiased under all random effects models and some mixed effects models with balanced data, for finding the explicit solution of maximum likelihood equations for the general mixed effects model and for showing the relationship between the spectral decomposition estimate and the analysis of variance estimate.     

Tail dependence between order statistics

In this work, we introduce the s,k-extremal coefficients for studying the tail dependence between the s-th lower and k-th upper order statistics of a normalized random vector. If its margins have tail dependence then so do their order statistics, with the strength of bivariate tail dependence decreasing as two order statistics become farther apart. Some general properties are derived for these dependence measures which can be expressed via copulas of random vectors. Its relations with other extremal dependence measures used in the literature are discussed, such as multivariate tail dependence coefficients, the coefficient η of tail dependence, coefficients based on tail dependence functions, the extremal coefficient ?, the multivariate extremal index and an extremal coefficient for min-stable distributions. Several examples are presented to illustrate the results, including multivariate exponential and multivariate Gumbel distributions widely used in applications.     

Dimension Free and Infinite Variance Tail Estimates on Poisson Space

Concentration inequalities are obtained on Poisson space, for random functionals with finite or infinite variance. In particular, dimension free tail estimates and exponential integrability results are given for the Euclidean norm of vectors of independent functionals. In the finite variance case these results are applied to infinitely divisible random variables such as quadratic Wiener functionals, including Lévy’s stochastic area and the square norm of Brownian paths. In the infinite variance case, various tail estimates such as stable ones are also presented.      

G-analysis of high-dimensional observations

Consistent and asymptotically normal G-estimators are obtained for generalized variance and normalized spectral function of the covariance matrix when Kolmogorov's condition is satisfied. The proofs are based on the application of limit theorems for random determinants and resolvents of random matrices.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 60, pp. 115–121, 1986.     

On spectral estimation for a homogeneous random process on the circle

A homogeneous random process on the circle {X(P): P∈S} is a process whose mean is zero and whose covariance function depends only on the angular distance θ between the points, i.e. E{X(P)}≡0 and E{X(P)X(Q)}=R(θ). We assume that the homogeneous process X(P) is observed at a finite number of points, equally spaced on the circle. Given independent realizations of the process, we first propose unbiased estimates for the parameters of the aliased spectrum and for the covariance function. We assume further that the process is Gaussian. The exact distribution of the spectral estimates and the asymptotic distribution of the estimates of the covariance function are derived. Finally, it is shown that the estimates proposed are in fact the maximum likelihood estimates and that they have minimum variance in the class of unbiased estimates.     

随机变量二次型的协方差在混合效应模型中的应用

本文提出方差分量ANOVA估计的一种改进方法, 证明了对于一般的方差分量模型, 只要方差分量的ANOVA估计存在就可以通过此方法给出其改进形式, 并且在均方误差意义下优于ANOVA估计. 特别地, 对于单向分类随机效应模型, Kelly和Mathew^[1]对ANOVA估计的改进就是我们提出的改进方法的特殊形式, 这也给出了此类改进估计在均方误差意义下优于ANOVA估计的另一种合理的解释. 同时, 本文又将此思想应用到对谱分解估计的改进上. 本文应用协方差的简单性质证明了对带有一个随机效应的方差分量模型, 当随机效应的协方差阵只有一个非零特征值时, 随机效应方差分量谱分解估计在均方误差意义下总是优于ANOVA估计. 本文最后将第三节的结论推广到广义谱分解估计下, 同时给出广义谱分解估计待定系数的一个合理的取值.