Zero-one laws for polynomials in Gaussian random variables: A simple proof

Zero-one laws for polynomials in Gaussian random variables have already been studied.⁽⁷⁾ They are established here by very simple arguments: Fubini's theorem and the rotational invariance of centered Gaussian measures. The proof is built on the Polarization formula that has received much attention in Refs. 1 and 5. Our point of view derives from the deep work of Borell.⁽²⁾ In a natural way, these results extend to finite-order Gaussian chaos processes.     

Gaussian series and daubechies operators

The subspace [Mtilde] of L²(Cⁿ) which is composed of Gaussian series and contains the subspace M spanned by Gaussian functions given in the paper [6] by Du and Wong has the proporety that the product of two Daubechies operators with symbols in [Mtilde] is a Daubechies operator with symbol H in [Mtilde]. Furthermore, an explicit expression for the symbol H is given     

The square of a Gaussian Markov process and nonlinear prediction

An explicit formula is obtained for the nonlinear predictor of Y(t) = X(t)² − E(X(t)²), where X(t) is an N-ple Gaussian Markov process.     

Non-linear prediction of the degree n of a Gaussian N-ple Markov process

An explicit formula is obtained for the nonlinear predicion of Y(t) = Xⁿ(t), where X(t) is an N-ple Gaussian Markov process.     

Functional Quantization and Small Ball Probabilities for Gaussian Processes

Quantization consists in studying the L ^r -error induced by the approximation of a random vector X by a vector (quantized version) taking a finite number n of values. We investigate this problem for Gaussian random vectors in an infinite dimensional Banach space and in particular, for Gaussian processes. A precise link proved by Fehringer⁽⁴⁾ and Dereich et al. ⁽³⁾ relates lower and upper bounds for small ball probabilities with upper and lower bounds for the quantization error, respectively. We establish a complete relationship by showing that the same holds for the direction from the quantization error to small ball probabilities. This allows us to compute the exact rate of convergence to zero of the minimal L ^r -quantization error from logarithmic small ball asymptotics and vice versa.     

A borderline Gaussian random Fourier series for the sample convergence in variation

We give an example of a Gaussian random Fourier series, of which the normalized remainders have their sojourn times converging in variation, namely the convergence in the space L¹(R) of the related density distributions, to the Gaussian density. The convergence in the space L^∞(R) is also proved. In our approach, we use local times of Gaussian random Fourier series.     

Recovering a Family of Two-Dimensional Gaussian Variables from the Minimum Process

Suppose that {(X _t, Y _t): t>}0 is a family of two independent Gaussian random variables with means m ₁(t) and m ₂(t) and variances σ ² ₁(t) and σ ² ₂(t). If at every time t>0 the first and second moment of the minimum process X _t∧Y _t are known, are the parameters governing these four moment functions uniquely determined ? We answer this question in the negative for a large class of Gaussian families including the “Brownian” case. Except for some degenerate situation where one variance function dominates the other, in which case the recovery of the parameters is fully successful, the second moment of the minimum process does not provide any additional clues on identifying the parameters.     

An Inversion-Free Estimating Equations Approach for Gaussian Process Models

One of the scalability bottlenecks for the large-scale usage of Gaussian processes is the computation of the maximum likelihood estimates of the parameters of the covariance matrix. The classical approach requires a Cholesky factorization of the dense covariance matrix for each optimization iteration. In this work, we present an estimating equations approach for the parameters of zero-mean Gaussian processes. The distinguishing feature of this approach is that no linear system needs to be solved with the covariance matrix. Our approach requires solving an optimization problem for which the main computational expense for the calculation of its objective and gradient is the evaluation of traces of products of the covariance matrix with itself and with its derivatives. For many problems, this is an O(nlog?n) effort, and it is always no larger than O(n²). We prove that when the covariance matrix has a bounded condition number, our approach has the same convergence rate as does maximum likelihood in that the Godambe information matrix of the resulting estimator is at least as large as a fixed fraction of the Fisher information matrix. We demonstrate the effectiveness of the proposed approach on two synthetic examples, one of which involves more than 1 million data points.     

Uniqueness of Embedding into a Gaussian Semigroup and a Poisson Semigroup with Determinate Jump Law on a Simply Connected Nilpotent Lie Group

Let {μ _t ⁽ⁱ⁾} _t≥0 (i=1,2) be continuous convolution semigroups (c.c.s.) on a simply connected nilpotent Lie group G. Suppose that μ ₁⁽¹⁾=μ ₁⁽²⁾. Assume furthermore that one of the following two conditions holds:

On Pseudo-Holomorphic Curves from Two-Spheres into a Complex Grassmannian G（2, 5）

Let s ： S2 → G（2, 5） be a linearly full totally unramified pseudo-holomorphic curve with constant Gaussian curvature K in a complex Grassmann manifold G（2, 5）. It is prove that K is either 1 4 1 or 4/5 if s is non-±holomorphic. Furthermore, K = 1/3 if and only if s is totally real. We also prove that the Gaussian curvature K is either 1 or -4/3 if s is a non-degenerate holomorphic curve under some conditions.     

Optimal transmission of “television type” gaussian signals through a noiseless feedback channel

Using the methods and results of the theory of conditionally Gaussian filtering of stochastic processes and fields, an optimal scheme of “television type” signal transmission through a noiseless feedback channel is constructed under the usual power conditions, the signals being evolutionary Gaussian fields θ={θ_t(x)),tε[0,T),xεDεRⁿ . Explicit representations for optimal coding and decoding functionals, which are optimal in the sense of a special square criterion, and the expression for the error of signal reproduction are given.     

Liminf results on the increments of an (N,d)-Gaussian process

We establish some liminf theorems on the increments of a (N,d)-Gaussian process with the usual Euclidean norm, via estimating upper bounds of large deviation probabilities on the suprema of the (N,d)-Gaussian process. This revised version was published online in June 2006 with corrections to the Cover Date.     

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.     

Asymptotic coefficients for Gaussian radial basis function interpolants

Gaussian radial basis functions (RBFs) on an infinite interval with uniform grid pacing h are defined by ?(x;α,h)≡exp(-[α²/h²]x²). The only significant numerical parameter is α, the inverse width of the RBF functions relative to h. In the limit α→0, we demonstrate that the coefficients of the interpolant of a typical function f(x) grow proportionally to exp(π²/[4α²]). However, we also show that the approximation to the constant f(x)≡1 is a Jacobian theta function whose coefficients do not blow up as α→0. The subtle interplay between the complex-plane singularities of f(x) (the function being approximated) and the RBF inverse width parameter α are analyzed. For α≈1/2, the size of the RBF coefficients and the condition number of the interpolation matrix are both no larger than O(10⁴) and the error saturation is smaller than machine epsilon, so this α is the center of a “safe operating range” for Gaussian RBFs.     

Convergence of Gaussian Quadrature Formulas for Power Orthogonal Polynomials

In classical theorems on the convergence of Gaussian quadrature formulas for power orthogonal polynomials with respect to a weight w on I =（a,b）,a function G ∈ S（w）:= { f:∫_I | f（x）| w（x）d x < ∞} satisfying the conditions G ^2j（x） ≥ 0,x ∈（a,b）,j = 0,1,...,and growing as fast as possible as x → a + and x → b,plays an important role.But to find such a function G is often difficult and complicated.This implies that to prove convergence of Gaussian quadrature formulas,it is enough to find a function G ∈ S（w） with G ≥ 0 satisfying sup n ∑λ0knG（xkn） k=1 n<∞ instead,where the xkn ’s are the zeros of the n th power orthogonal polynomial with respect to the weight w and λ0kn ’s are the corresponding Cotes numbers.Furthermore,some results of the convergence for Gaussian quadrature formulas involving the above condition are given.     

A Note on Transient Gaussian Fluid Models

In this paper we study the distribution of the supremum over interval [0,T] of a centered Gaussian process with stationary increments with a general negative drift function. This problem is related to the distribution of the buffer content in a transient Gaussian fluid queue Q(T) at time T, provided that at time 0 the buffer is empty. The general theory is illustrated by detailed considerations of different cases for the integrated Gaussian process and the fractional Brownian motion. We give asymptotic results for P(Q(T)>x) and P(sup _0tT Q(t)>x) as x.     

Central limit theorem for first-passage percolation time across thin cylinders

We prove that first-passage percolation times across thin cylinders of the form [0, n] × [?h _n , h _n ] ^d-1 obey Gaussian central limit theorems as long as h _n grows slower than n ^1/(d+1). It is an open question as to what is the fastest that h _n can grow so that a Gaussian CLT still holds. Under the natural but unproven assumption about existence of fluctuation and transversal exponents, and strict convexity of the limiting shape in the direction of (1, 0, . . . , 0), we prove that in dimensions 2 and 3 the CLT holds all the way up to the height of the unrestricted geodesic. We also provide some numerical evidence in support of the conjecture in dimension 2.     

Inequalities for the Gaussian hypergeometric function

we study the monotonicity of certain combinations of the Gaussian hypergeometric functions F(-1/2,1/2;1;1- xc) and F(-1/2- δ,1/2 + δ;1;1- xd) on(0,1) for given 0 c 5d/6 ∞ andδ∈(-1/2,1/2),and find the largest value δ1 = δ1(c,d) such that inequality F(-1/2,1/2;1;1- xc) F(-1/2- δ,1/2 + δ;1;1- xd) holds for all x ∈(0,1). Besides,we also consider the Gaussian hypergeometric functions F(a- 1- δ,1- a + δ;1;1- x3) and F(a- 1,1- a;1;1- x2) for given a ∈ [1/29,1) and δ∈(a- 1,a),and obtain the analogous results.     

Gaussian Polynomials and Content Ideal in Pullbacks

This article deals mainly with rings (with zerodivisors) in which regular Gaussian polynomials have locally principal contents. Precisely, we show that if (T,M) is a local ring which is not a field, D is a subring of T/M such that qf(D) = T/M, h: T → T/M is the canonical surjection and R = h ^?1(D), then if T satisfies the property “every regular Gaussian polynomial has locally principal content,” then also R verifies the same property. We also show that if D is a Prüfer domain and T satisfies the property “every Gaussian polynomial has locally principal content”, then R satisfies the same property. The article includes a brief discussion of the scopes and limits of our result.     

Excursions of a Gaussian process with variable variance above a barrier increasing to infinity

For a family of real-valued Gaussian processes ξ _u (t), t ∈ [0, T], we obtain an exact asymptotics of the probability of crossing a level u as u → ∞ under certain conditions on the variance and correlation. This result is applied to the investigation of excursions of a stationary zero-mean process above a barrier increasing to infinity.