Centered Gaussian Process Research Articles

A Gaussian limit process for optimal FIND algorithms

We consider versions of the FIND algorithm where the pivot element used is the median of a subset chosen uniformly at random from the data. For the median selection we assume that subsamples of size asymptotic to $c \cdot n^\alpha$ are chosen, where $0 < \alpha \leq \frac{1}{2}$, $c > 0$ and $n$ is the size of the data set to be split. We consider the complexity of FIND as a process in the rank to be selected and measured by the number of key comparisons required. After normalization we show weak convergence of the complexity to a centered Gaussian process as $n \to \infty$, which depends on $\alpha$. The proof relies on a contraction argument for probability distributions on càdlàg functions. We also identify the covariance function of the Gaussian limit process and discuss path and tail properties.

Testing for the buffered autoregressive processes

This paper investigates a quasi-likelihood ratio (LR) test for the thresh- olds in buffered autoregressive processes. Under the null hypothesis of no threshold, the LR test statistic converges to a function of a centered Gaussian process. Un- der local alternatives, this LR test has nontrivial asymptotic power. A bootstrap method is proposed to obtain the critical value for the LR test. Simulation studies and an example are given to assess the performance of the test. The proof here is not standard and can be used in other non-linear time series models.

Integrated conditional moment test for partially linear single index models incorporating dimension-reduction

Studying model checking problems for partially linear single-index models, we propose a variant of the integrated conditional moment test using a linear projection weighting function, which gains dimension reduction and makes the proposed method act as if there exists only one covariate even in the presence of multiple dimensional regressors. We derive asymptotic distributions of the proposed test; i.e., an integral of a centered Gaussian process under the null hypothesis and an integral of a non-centered one under Pitman local alternatives. We also suggest a consistent bootstrap procedure for calculating the critical values. Simulation studies are conducted to demonstrate the performance of the proposed procedure and a real example is analyzed for an illustration.

LIGHT-CONE FLUCTUATIONS AND THE RENORMALIZED STRESS TENSOR OF A MASSLESS SCALAR FIELD

We investigate the effects of light-cone fluctuations over the renormalized vacuum expectation value of the stress–energy tensor of a real massless minimally coupled scalar field defined in a (d+1)-dimensional flat space–time with topology [Formula: see text]. For modeling the influence of light-cone fluctuations over the quantum field, we consider a random Klein–Gordon equation. We study the case of centered Gaussian processes. After taking into account all the realizations of the random processes, we present the correction caused by random fluctuations. The averaged renormalized vacuum expectation value of the stress–energy associated with the scalar field is presented.

Laws of Large Numbers and Langevin Approximations for Stochastic Neural Field Equations

In this study, we consider limit theorems for microscopic stochastic models of neural fields. We show that the Wilson–Cowan equation can be obtained as the limit in uniform convergence on compacts in probability for a sequence of microscopic models when the number of neuron populations distributed in space and the number of neurons per population tend to infinity. This result also allows to obtain limits for qualitatively different stochastic convergence concepts, e.g., convergence in the mean. Further, we present a central limit theorem for the martingale part of the microscopic models which, suitably re-scaled, converges to a centred Gaussian process with independent increments. These two results provide the basis for presenting the neural field Langevin equation, a stochastic differential equation taking values in a Hilbert space, which is the infinite-dimensional analogue of the chemical Langevin equation in the present setting. On a technical level, we apply recently developed law of large numbers and central limit theorems for piecewise deterministic processes taking values in Hilbert spaces to a master equation formulation of stochastic neuronal network models. These theorems are valid for processes taking values in Hilbert spaces, and by this are able to incorporate spatial structures of the underlying model.Mathematics Subject Classification (2000): 60F05, 60J25, 60J75, 92C20.

Significance testing in quantile regression

We consider the problem of testing significance of predictors in multivariate nonparametric quantile regression. A stochastic process is proposed, which is based on a comparison of the responses with a nonparametric quantile regression estimate under the null hypothesis. It is demonstrated that under the null hypothesis this process converges weakly to a centered Gaussian process and the asymptotic properties of the test under fixed and local alternatives are also discussed. In particular we show, that - in contrast to the nonparametric approach based on estimation of $L^{2}$-distances - the new test is able to detect local alternatives which converge to the null hypothesis with any rate $a_{n}\to 0$ such that $a_{n}\sqrt{n}\to\infty$ (here $n$ denotes the sample size). We also present a small simulation study illustrating the finite sample properties of a bootstrap version of the corresponding Kolmogorov-Smirnov test.

Asymptotic behavior of the convex hull of a stationary Gaussian process∗

Let X = {X(t), t ∈ T} be a stationary centered Gaussian process with values in ℝ d , where the parameter set T equals ℕ or ℝ+. Let Σ t = Cov(X 0 ,X t ) be the covariance function of X, and (Ω,ℱ, P) be the underlying probability space. We consider the asymptotic behavior of convex hulls W t = conv{X u , u ∈ T ∩ [0, t]} as t → +∞ and show that under the condition Σt → 0, t→∞, the rescaled convex hull (2 ln t) −1/2 W t converges almost surely (in the sense of Hausdorff distance) to an ellipsoid ℰ associated to the covariance matrix Σ 0. The asymptotic behavior of the mathematical expectations E f(W t ), where f is a homogeneous function, is also studied. These results complement and generalize in some sense the results of Davydov [Y. Davydov, On convex hull of Gaussian samples, Lith. Math. J., 51(2): 171–179, 2011].

Continuous Gaussian Multifractional Processes with Random Pointwise Hölder Regularity

Let {X(t)} t∈ℝ be an arbitrary centered Gaussian process whose trajectories are, with probability 1, continuous nowhere differentiable functions. It follows from a classical result, derived from zero-one law, that, with probability 1, the trajectories of X have the same global Hölder regularity over any compact interval, i.e. the uniform Hölder exponent does not depend on the choice of a trajectory. A similar phenomenon occurs with their local Hölder regularity measured through the local Hölder exponent. Therefore, it seems natural to ask the following question: Does such a phenomenon also occur with their pointwise Hölder regularity measured through the pointwise Hölder exponent?In this article, using the framework of multifractional processes, we construct a family of counterexamples showing that the answer to this question is not always positive.

Gaussian queues in light and heavy traffic

In this paper we investigate Gaussian queues in the light-traffic and in the heavy-traffic regime. Let $Q^{(c)}_{X}\equiv\{Q^{(c)}_{X}(t):t\ge0\}$ denote a stationary buffer content process for a fluid queue fed by the centered Gaussian process X≡{X(t):t∈ℝ} with stationary increments, X(0)=0, continuous sample paths and variance function σ 2(⋅). The system is drained at a constant rate c>0, so that for any t≥0, We study $Q^{(c)}_{X}\equiv\{Q_{X}^{(c)}(t):t\ge0\}$ in the regimes c→0 (heavy traffic) and c→∞ (light traffic). We show for both limiting regimes that, under mild regularity conditions on σ, there exists a normalizing function δ(c) such that $Q^{(c)}_{X}(\delta(c)\cdot)/\sigma(\delta(c))$ converges to $Q^{(1)}_{B_{H}}(\cdot)$ in C[0,∞), where B H is a fractional Brownian motion with suitably chosen Hurst parameter H.

Exact asymptotics of supremum of a stationary Gaussian process over a random interval

Let { X ( t ) : t ∈ [ 0 , ∞ ) } be a centered stationary Gaussian process. We study the exact asymptotics of P ( sup s ∈ [ 0 , T ] X ( s ) > u ) , as u → ∞ , where T is an independent of { X ( t ) } nonnegative random variable. It appears that the heaviness of T impacts the form of the asymptotics, leading to three scenarios: the case of integrable T , the case of T having regularly varying tail distribution with parameter λ ∈ ( 0 , 1 ) and the case of T having slowly varying tail distribution.

Scalar quantum field theory in disordered media

A free massive scalar field in inhomogeneous random media is investigated. The coefficients of the Klein-Gordon equation are taken to be random functions of the spatial coordinates. The case of an annealed-like disordered medium, modeled by centered stationary and Gaussian processes, is analyzed. After performing the averages over the random functions, we obtain the two-point causal Green's function of the model up to one-loop. The disordered scalar quantum field theory becomes qualitatively similar to a $\lambda\phi^{4}$ self-interacting theory with a frequency-dependent coupling.

Extremes of the time-average of stationary Gaussian processes

We study the exact asymptotics of P ( sup t ≥ 0 I Z ( t ) > u ) , as u → ∞ , where I Z ( t ) = { 1 t ∫ 0 t Z ( s ) d s for t > 0 Z ( 0 ) for t = 0 and { Z ( t ) : t ≥ 0 } is a centered stationary Gaussian process with covariance function satisfying some regularity conditions. As an application, we analyze the probability of buffer emptiness in a Gaussian fluid queueing system and the collision probability of differentiable Gaussian processes with stationary increments. Additionally, we find estimates for analogues of Piterbarg–Prisyazhnyuk constants, that appear in the form of the considered asymptotics.

Lévy area for Gaussian processes: A double Wiener–Itô integral approach

Let { X 1 ( t ) } 0 ≤ t ≤ 1 and { X 2 ( t ) } 0 ≤ t ≤ 1 be two independent continuous centered Gaussian processes with covariance functions R 1 and R 2 . We show that if the covariance functions are of finite p -variation and q -variation respectively and such that p − 1 + q − 1 > 1 , then the Lévy area can be defined as a double Wiener–Itô integral with respect to an isonormal Gaussian process induced by X 1 and X 2 . Moreover, some properties of the characteristic function of that generalised Lévy area are studied.

On convex hull of Gaussian samples

Let X i = {X i (t), t ∈ T} be i.i.d. copies of a centered Gaussian process X = {X(t), t ∈ T} with values in $ {\mathbb{R}^d} $ defined on a separable metric space T. It is supposed that X is bounded. We consider the asymptotic behavior of convex hulls $$ {W_n} = {\text{conv}}\left\{ {{X_1}(t), \ldots, {X_n}(t),\,\,t \in T} \right\} $$ and show that, with probability 1, $$ \mathop {{\lim }}\limits_{n \to \infty } \frac{1}{{\sqrt {{2\ln n}} }}{W_n} = W $$ (in the sense of Hausdorff distance), where the limit shape W is defined by the covariance structure of X: W = conv{K t , t ∈ T}, Kt being the concentration ellipsoid of X(t). We also study the asymptotic behavior of the mathematical expectations E f(W n ), where f is an homogeneous functional.

Asymptotics of supremum distribution of a Gaussian process over a Weibullian time

Let $\{X(t) :t∈[0, ∞)\}$ be a centered Gaussian process with stationary increments and variance function $σ_X^2(t)$. We study the exact asymptotics of $ℙ(\sup _{t∈[0, T]}X(t)>u)$ as $u→∞$, where $T$ is an independent of $\{X(t)\}$ non-negative Weibullian random variable. As an illustration, we work out the asymptotics of the supremum distribution of fractional Laplace motion.

Random Gaussian Sums on Trees

Let $T$ be a tree with induced partial order. We investigate a centered Gaussian process $X$ indexed by $T$ and generated by weight functions. In a first part we treat general trees and weights and derive necessary and sufficient conditions for the a.s. boundedness of $X$ in terms of compactness properties of $(T,d)$. Here $d$ is a special metric defined by the weights, which, in general, is not comparable with the Dudley metric generated by $X$. In a second part we investigate the boundedness of $X$ for the binary tree. Assuming some mild regularity assumptions about on weight, we completely characterize homogeneous weights with $X$ being a.s. bounded.

Winding of planar Gaussian processes

We consider a smooth, rotationally invariant, centered Gaussianprocess in the plane, with arbitrary correlation matrixCtt′. We study thewinding angle ϕt, around its center. We obtain a closed formula for the variance of the winding angle as a function of the matrixCtt′. For moststationary processes Ctt′ = C(t−t′) the winding angle exhibits diffusion at large time with diffusion coefficient . Correlations of exp(inϕt) with integer n, the distribution of the angular velocity , and the variance of the algebraic area are also obtained. For smooth processes withstationary increments (random walks) the variance of the winding angle grows as , with proper generalizations to the various classes of fractionalBrownian motion. These results are tested numerically. Non-integern is studied numerically.

On the Traces of Laguerre Processes

Almost sure and $L^k$-convergence of the traces of Laguerre processes to the family of dilations of the standard free Poisson distribution are established. We also prove that the fluctuations around the limiting process, converge weakly to a continuous centered Gaussian process. The almost sure convergence on compact time intervals of the largest and smallest eigenvalues processes is also established

A Cox Process Involved in the Bose–Einstein Condensation

The point process corresponding to the configurations of bosons in standard conditions is a Cox process driven by the square norm of a centered Gaussian process. This point process is infinitely divisible. We point out the fact that this property is preserved by the Bose–Einstein condensation phenomenon and show that the obtained point process after such a condensation occured, is still a Cox process but driven by the square norm of a shifted Gaussian process, the shift depending on the density of the particles. This law provides an illustration of a “super”- Isomorphism Theorem existing above the usual Isomorphism Theorem of Dynkin available for Gaussian processes.

On permanental processes

Permanental processes can be viewed as a generalization of squared centered Gaussian processes. We analyze the connections of these processes with the local time process of general Markov processes. The obtained results are related to the notion of infinite divisibility.