Correlation Inequalities and Applications to Vector-Valued Gaussian Random Variables and Fractional Brownian Motion

In this paper we extend certain correlation inequalities for vector-valued Gaussian random variables due to Kolmogorov and Rozanov. The inequalities are applied to sequences of Gaussian random variables and Gaussian processes. For sequences of Gaussian random variables satisfying a correlation assumption, we prove a Borel-Cantelli lemma, maximal inequalities and several laws of large numbers. This extends results of Be?ka and Ciesielski and of Hytönen and the author. In the second part of the paper we consider a certain class of vector-valued Gaussian processes which are α-Hölder continuous in p-th moment. For these processes we obtain Besov regularity of the paths of order α. We also obtain estimates for the moments in the Besov norm. In particular, the results are applied to vector-valued fractional Brownian motions. These results extend earlier work of Ciesielski, Kerkyacharian and Roynette and of Hytönen and the author.     

Oscillation of Gaussian processes

The properties of the oscillations of Banach-valued Gaussian processes are investigated. The oscillations of several Gaussian sequences are computed. The obtained results are used for the investigation of the properties of the trajectories of one-dimensional Gaussian processes.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 177, pp. 92–97, 1989.     

Permanental vectors

A permanental vector is a generalization of a vector with components that are squares of the components of a Gaussian vector, in the sense that the matrix that appears in the Laplace transform of the vector of Gaussian squares is not required to be either symmetric or positive definite. In addition, the power of the determinant in the Laplace transform of the vector of Gaussian squares, which is −1/2, is allowed to be any number less than zero.     

Central limit theorem for constrained Poisson systems

We prove that for a class of constrained Poisson white noise fields, the scaling (continuum) limit exists and equals Gaussian white noise, indexed by mean zero test functions. Under natural conditions on the Lévy measure, the (Poisson) moments converge to their Gaussian counterparts.     

On partial sums of a gaussian sequence with stationary increments

We establish large increment properties for a Gaussian sequence with stationary increments under global conditions. Limit theorems for partial sums of the sequence with nonpositive correlation functions or the correlation functions on their speed of convergence to zero are proved via estimating a probability inequality on the supremum of Gaussian processes     

Lattice Points in Three-Dimensional Convex Bodies with Points of Gaussian Curvature Zero at the Boundary

 In this article we investigate the number of lattice points in a three-dimensional convex body which contains non-isolated points with Gaussian curvature zero but a finite number of flat points at the boundary. Especially, in case of rational tangential planes in these points we investigate not only the influence of the flat points but also of the other points with Gaussian curvature zero on the estimation of the lattice rest.     

Upper functions and oscillating Gaussian processes

We study the behavior of unbounded Gaussian processes. We give necessary and sufficient conditions for the existence of exact upper sequences for unbounded Gaussian sequences with bounded variances.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 158, pp. 5–13, 1987.     

On Global and Local Properties of the Trajectories of Gaussian Random Fields——A Look Through the Set of Limit Points

This paper studies the global and local properties of the trajectories of Gaussian random fields with stationary increments and proves sufficient conditions for Strassen's functional laws of the iterated logarithm at zero and infinity respectively.The sets of limit points of those Gaussian random fields are obtained.The main results are applied to fractional Riesz-Bessel processes and the sets of limit points of this field are obtained.     

The Asymptotic Distribution of REML Estimators

Restricted maximum likelihood (REML) estimation is a method employed to estimate variance-covariance parameters from data that follow a Gaussian linear model. In applications, it has either been conjectured or assumed that REML estimators are asymptotically Gaussian with zero mean and variance matrix equal to the inverse of the restricted information matrix. In this article, we give conditions under which the conjecture is true and apply our results to variance-components models. An important application of variance components is to census undercount; a simulation is carried out to verify REML′s properties for a typical census undercount model.     

Strong approximation for sums of asymptotically negatively dependent Gaussian sequences with applications

By applying the Skorohod martingale embedding method, a strong approximation theorem for partial sums of asymptotically negatively dependent (AND) Gaussian sequences, under polynomial decay rates, is established. As applications, the law of the iterated logarithm, the Chung-type law of the iterated logarithm and the almost sure central limit theorem for AND Gaussian sequences are derived.     

Large deviation properties for patterns

Deciding whether a given pattern is over- or under-represented according to a given background model is a key question in computational biology. Such a decision is usually made by computing some p-values reflecting the “exceptionality” of a pattern in a given sequence or set of sequences. In the simplest cases (short and simple patterns, simple background model, small number of sequences), an exact p-value can be computed with a tractable complexity. The realistic cases are in general too complicated to get such an exact p-value. Approximations are thus proposed (Gaussian, Poisson, Large deviation approximations). These approximations are applicable under some conditions: Gaussian approximations are valid in the central domain while Poisson and Large deviation approximations are valid for rare events. In the present paper, we prove a large deviation approximation to the double strands counting problem that refers to a counting of a given pattern in a set of sequences that arise from both strands of the genome. In that case, dependencies between a sequence and its reverse complement cannot be neglected. They are captured here for a Bernoulli model from general combinatorial properties of the pattern. A large deviation result is also provided for a set of small sequences.     

Lattice Points in Three-Dimensional Convex Bodies with Points of Gaussian Curvature Zero at the Boundary

 In this article we investigate the number of lattice points in a three-dimensional convex body which contains non-isolated points with Gaussian curvature zero but a finite number of flat points at the boundary. Especially, in case of rational tangential planes in these points we investigate not only the influence of the flat points but also of the other points with Gaussian curvature zero on the estimation of the lattice rest. Received 19 June 2001; in revised form 17 January 2002 RID="a" ID="a" Dedicated to Professor Edmund Hlawka on the occasion of his 85th birthday     

Positivity for Gaussian graphical models

Gaussian graphical models are parametric statistical models for jointly normal random variables whose dependence structure is determined by a graph. In previous work, we introduced trek separation, which gives a necessary and sufficient condition in terms of the graph for when a subdeterminant is zero for all covariance matrices that belong to the Gaussian graphical model. Here we extend this result to give explicit cancellation-free formulas for the expansions of non-zero subdeterminants.     

On Piterbarg theorem for maxima of stationary Gaussian sequences

Limit distributions of maxima of dependent Gaussian sequence are different according to the convergence rate of their correlations. For three different conditions on convergence rate of the correlations, in this paper, we establish the Piterbarg theorem for maxima of stationary Gaussian sequences.     

Departure from normality of increasing-dimension martingales

In this paper, we consider sequences of vector martingale differences of increasing dimension. We show that the Kantorovich distance from the distribution of the k(n)-dimensional average of n martingale differences to the corresponding Gaussian distribution satisfies certain inequalities. As a consequence, if the growth of k(n) is not too fast, then the Kantorovich distance converges to zero. Two applications of this result are presented. The first is a precise proof of the asymptotic distribution of the multivariate portmanteau statistic applied to the residuals of an autoregressive model and the second is a proof of the asymptotic normality of the estimates of a finite autoregressive model when the process is an AR(∞) and the order of the model grows with the length of the series.     

Score tests for inverse Gaussian mixtures

The mixed inverse Gaussian given by Whitmore (biScand. J. Statist., 13 , 1986, 211–220) provides a convenient way for testing the goodness‐of‐fit of a pure inverse Gaussian distribution. The test is a one‐sided score test with the null hypothesis being the pure inverse Gaussian (i.e. the mixing parameter is zero) and the alternative a mixture. We devise a simple score test and study its finite sample properties. Monte Carlo results show that it compares favourably with the smooth test of Ducharme ( Test , 10 , 2001, 271‐290). In practical applications, when the pure inverse Gaussian distribution is rejected, one is interested in making inference about the general values of the mixing parameter. However, as it is well known that the inverse Gaussian mixture is a defective distribution; hence, the standard likelihood inference cannot be applied. We propose several alternatives and provide score tests for the mixing parameter. Finite sample properties of these tests are examined by Monte Carlo simulation. Copyright © 2010 John Wiley & Sons, Ltd.     

On approximating the probability of a large excursion of a nonstationary Gaussian process

We consider a nonstationary Gaussian process with the zero mean and unit variance which possesses the mean square derivative. We study the asymptotic behavior of the maximum Gaussian processes on both finite and increasing intervals. The results are applied to studying the maximal deviation of empirical density and the regression curve on a finite interval.     

Limit theorems for Fourier transforms of functionals of Gaussian sequences

Summary Limit theorems with a non-Gaussian (in fact nonstable) limiting distribution have been obtained under suitable conditions for partial sums of instantaneous nonlinear functions of stationary Gaussian sequences with long range dependence. Analogous limit theorems are here obtained for finite Fourier transforms of instantaneous nonlinear functions of stationary Gaussian sequences with long range dependence.Research supported in part by the Office of Naval Research Contract N00014-75C-0428.     

Stabilization of statistics in wave and Klein-Gordon equations with mixing. Scattering theory for infinite energy solutions

We consider wave and Klein-Gordon equations in the whole space ?ⁿ with arbitraryn≥2. We assume initial data to be homogeneous random functions in ?ⁿ with zero expectation and finite mean density of energy. Moreover, we assume initial data fit mixing condition of Ibragimov-Linnik type. We consider the distributions of the random solution at the moment of timet. The main results mean the convergence of this distribution to some Gaussian measure ast→∞. This is a central limit theorem for wave and Klein-Gordon equations. The limit Gaussian measures are invariant measures for equations considered. Corresponding stationary random solutions are ergodic and mixing in time. The results are inspired by mathematical problems of statistical physics.     

On hypersurfaces in a Riemannian vector bundle with prescribed Gaussian curvature and convexity

Let M be a compact Riemannian manifold, E a Riemannian vector bundle on M and the unit subbundle of E. We prove that there is no radial graph on with a strictly positive Gaussian curvature and that the Gaussian curvature of a convex radial graph must be identically equal to zero. Moreover, by solving on a nonlinear degenerate equation of Monge-Ampère type, we prove the existence of radial graphs having simultaneously a Gaussian curvature identically equal to zero and a prescribed strictly positive vertical Gaussian curvature.Mathematics Subject Classification (2000):35J60, 53C55, 58G30