Class Of Gaussian Processes Research Articles

Harmonic analysis meets stationarity: A general framework for series expansions of special Gaussian processes

In this paper, we present a new approach to derive series expansions for some Gaussian processes based on harmonic analysis of their covariance function. In particular, we propose a new simple rate-optimal series expansion for fractional Brownian motion. The convergence of the latter series holds in mean square and uniformly almost surely, with a rate-optimal decay of the remainder of the series. We also develop a general framework of convergent series expansions for certain classes of Gaussian processes with stationarity. Finally, an application to optimal functional quantization is described.

Asymptotic behavior of the weak approximation to a class of Gaussian processes

AbstractWe study, under mild conditions, the weak approximation constructed from a standard Poisson process for a class of Gaussian processes, and establish its sample path moderate deviations. The techniques consist of a good asymptotic exponential approximation in moderate deviations, the Besov–Lèvy modulus embedding, and an exponential martingale technique. Moreover, our results are applied to the weak approximations associated with the moving average of Brownian motion, fractional Brownian motion, and an Ornstein–Uhlenbeck process.

Sequential Learning of Active Subspaces

ABSTRACT In recent years, active subspace methods (ASMs) have become a popular means of performing subspace sensitivity analysis on black-box functions. Naively applied, however, ASMs require gradient evaluations of the target function. In the event of noisy, expensive, or stochastic simulators, evaluating gradients via finite differencing may be infeasible. In such cases, often a surrogate model is employed, on which finite differencing is performed. When the surrogate model is a Gaussian process (GP), we show that the ASM estimator is available in closed form, rendering the finite-difference approximation unnecessary. We use our closed-form solution to develop acquisition functions focused on sequential learning tailored to sensitivity analysis on top of ASMs. We also show that the traditional ASM estimator may be viewed as a method of moments estimator for a certain class of GPs. We demonstrate how uncertainty on GP hyperparameters may be propagated to uncertainty on the sensitivity analysis, allowing model-based confidence intervals on the active subspace. Our methodological developments are illustrated on several examples. Supplementary files for this article are available online.

The principle of invariance in the Donsker form to the partial sum processes of finite order moving averages

We consider the process of partial sums of moving averages of finite order with a regular varying memory function, constructed from a stationary sequence, variance of the sum of which is a regularly varying function. We study the Gaussian approximation of this process of partial sums with the aid of a certain class of Gaussian processes, and obtain sufficient conditions for the $C$-convergence in the invariance principle in the Donsker form

Limit theorems for Betti numbers of extreme sample clouds with application to persistence barcodes

We investigate the topological dynamics of extreme sample clouds generated by a heavy tail distribution on $\mathbb{R}^{d}$ by establishing various limit theorems for Betti numbers, a basic quantifier of algebraic topology. It then turns out that the growth rate of the Betti numbers and the properties of the limiting processes all depend on the distance of the region of interest from the weak core, that is, the area in which random points are placed sufficiently densely to connect with one another. If the region of interest becomes sufficiently close to the weak core, the limiting process involves a new class of Gaussian processes. We also derive the limit theorems for the sum of bar lengths in the persistence barcode plot, a graphical descriptor of persistent homology.

Clark formula for local time for one class of Gaussian processes

In the article we present chaotic decomposition and analog of the Clark formula for the local time of Gaussian integrators. Since the integral with respect to Gaussian integrator is understood in Skorokhod sense, then there exist more than one Clark representation for the local time. We present different representations and discuss the representation with the minimal L_2-norm.

Properties of Gaussian local times

We investigate properties of local time for one class of Gaussian processes. These processes are called integrators since every function from L2([0; 1]) can be integrated over it. Using the white noise representation, we can associate integrators with continuous linear operators in L2([0; 1]). In terms of these operators, we discuss the existence and properties of local time for integrators. Also, we study the asymptotic behavior of Brownian self-intersection local time as its end-point tends to infinity.

Persistence Probability for a Class of Gaussian Processes Related to Random Interface Models

We consider a class of Gaussian processes which are obtained as height processes of some (d+ 1)-dimensional dynamic random interface model on ℤd. We give an estimate of persistence probability, namely, largeTasymptotics of the probability that the process does not exceed a fixed level up to timeT. The interaction of the model affects the persistence probability and its asymptotics changes depending on the dimensiond.

Persistence Probability for a Class of Gaussian Processes Related to Random Interface Models

We consider a class of Gaussian processes which are obtained as height processes of some (d+ 1)-dimensional dynamic random interface model on ℤd. We give an estimate of persistence probability, namely, largeTasymptotics of the probability that the process does not exceed a fixed level up to timeT. The interaction of the model affects the persistence probability and its asymptotics changes depending on the dimensiond.

A central limit theorem for a weighted power variation of a Gaussian process*

Let ρ be a real-valued function on [0, T], and let LSI(ρ) be a class of Gaussian processes over time interval [0, T], which need not have stationary increments but their incremental variance σ(s, t) is close to the values ρ(|t − s|) as t → s uniformly in s ∈ (0, T]. For a Gaussian processesGfrom LSI(ρ), we consider a power variation V n corresponding to a regular partition π n of [0, T] and weighted by values of ρ(·). Under suitable hypotheses on G, we prove that a central limit theorem holds for V n as the mesh of π n approaches zero. The proof is based on a general central limit theorem for random variables that admit a Wiener chaos representation. The present result extends the central limit theorem for a power variation of a class of Gaussian processes with stationary increments and for bifractional and subfractional Gaussian processes.

A Hull and White formula for a stochastic volatility Lévy model with infinite activity

The new class of Gaussian processes which are obtained by a compact perturbation in the reproducing kernel of Wiener process is in- troduced. The nite families of increments of our processes on small time intervals behave as increments of Wiener process. Consequently a lot of asymptotical properties of Wiener process are inherited. The law of iterated logarithm, the analogue of the Levy modulus of continuity and almost unifor- mity of hitting distribution on the small circles are proved. The renormalized Fourier{Wiener transform of the self-intersection local time for compactly perturbed Wiener process is constructed. In this article we consider self-intersection local times for one class of planar Gaussian processes. The interest to the self-intersection local times of planar Brownian motion has a long history since the theorem of Dvoretzky, Erdos, Kaku- tani (3) which established the existence of multiple self-intersections. The various kinds of renormalization were proposed for the self-intersection local times of pla- nar Brownian motion and certain Levy processes in the articles (4, 5, 10, 15, 16). Most of these articles essentially use the Markov property of the considered pro- cess. The aim of this paper is to present an approach to investigation of planar Gaussian processes which does not use the Markov property. We introduce a new class of Gaussian processes which are obtained with the help of compact pertur- bation in the reproducing kernel of Wiener process. Such processes inherit many properties of Wiener process. As an example we prove here the law of iterated logarithm, the analogue of the Levy modulus of continuity and almost uniformity of hitting distribution on the small circles. The nite families of small increments of our processes behave like the increments of Wiener process. This allows us to prove that compactly perturbed Wiener process has strong local nondeterminism property, which is a generalization of local nondeterminism property introduced by S.Berman (1). Finally we present the renormalization for Fourier{Wiener trans- form of the self-intersection local times for compactly perturbed Wiener process.

Spatial prediction with mobile sensor networks using Gaussian processes with built-in Gaussian Markov random fields

In this paper, a new class of Gaussian processes is proposed for resource-constrained mobile sensor networks. Such a Gaussian process builds on a GMRF with respect to a proximity graph over a surveillance region. The main advantages of using this class of Gaussian processes over standard Gaussian processes defined by mean and covariance functions are its numerical efficiency and scalability due to its built-in GMRF and its capability of representing a wide range of non-stationary physical processes. The formulas for predictive statistics such as predictive mean and variance are derived and a sequential field prediction algorithm is provided for sequentially sampled observations. For a special case using compactly supported weighting functions, we propose a distributed algorithm to implement field prediction by correctly fusing all observations. Simulation and experimental results illustrate the effectiveness of our approach.

White noise based stochastic calculus associated with a class of Gaussian processes

Using the white noise space setting, we define and study stochastic integrals with respect to a class of stationary increment Gaussian processes. We focus mainly on continuous functions with values in the Kondratiev space of stochastic distributions, where use is made of the topology of nuclear spaces. We also prove an associated Ito formula.

Limit theorems for a quadratic variation of Gaussian processes

In the paper a weighted quadratic variation based on a sequence of partitions for a class of Gaussian processes is considered. Conditions on the sequence of partitions and the process are established for the quadratic variation to converge almost surely and for a central limit theorem to be true. Also applications to bifractional and sub-fractional Brownian motion and the estimation of their parameters are provided.

Degenerate self-similar measures, spectral asymptotics and small deviations of Gaussian processes

We find the logarithmic small ball asymptotics for the L2-norm with respect to degenerate self-similar measures of a certain class of Gaussian processes, including Brownian motion, Ornstein–Uhlenbeck process and their integrated counterparts.

A class of Gaussian processes with fractional spectral measures

We study a family of stationary increment Gaussian processes, indexed by time. These processes are determined by certain measures σ (generalized spectral measures), and our focus here is on the case when the measure σ is a singular measure. We characterize the processes arising from σ when σ is in one of the classes of affine selfsimilar measures. Our analysis makes use of Kondratiev white noise spaces. With the use of a priori estimates and the Wick calculus, we extend and sharpen (see Theorem 7.1) earlier computations of Ito stochastic integration developed for the special case of stationary increment processes having absolutely continuous measures. We further obtain an associated Ito formula (see Theorem 8.1).

Spacings-ratio empirical processes

We consider an empirical process based upon ratios of selected pairs of spacings, generated by independent samples of arbitrary sizes. As a main result, we show that when both samples are uniformly distributed on (possibly shifted) intervals of equal lengths, this empirical process converges to a mean-centered Brownian bridge of the form B C (u) = B(u)−6Cu(1−u) Σ 0 1 B(s)ds, where B(·) denotes a Brownian bridge, and C, a constant. The investigation of the class of Gaussian processes {B C (·): C ∈ ℝ} leads to some unexpected distributional identities such as B 2(·) $$ \underline{\underline d} $$ B(·). We discuss this and similar results in an extended framework.

A Class of Gaussian Hybrid Processes for Modeling Financial Markets

This paper proposes a one-factor model of financial markets using a class of Gaussian process that can be decomposed into a Brownian motion and an Ornstein–Uhlenbeck process. It is shown that this “hybrid” process is obtained as a continuous-time scaling limit of the differenced first-order autoregressive integrated moving average (ARIMA(1,1,1)) process. Parameter estimations using an ARIMA(1,1,1) framework and its variance ratio test show the accuracy of the proposed model. Construction of the one-factor commodity futures price model is presented as an application. A multidimensional extension of the hybrid process is also presented in the Appendix.

Sharp Small Deviation Asymptotics in the L 2-Norm for a Class of Gaussian Processes

We characterize the exact behavior of small deviations in a Hilbert norm for centered Gaussian processes in the case where their covariances have a special form of eigenvalues. This result enables us to describe small deviation asymptotics for certain Gaussian processes. Bibliography: 20 titles.

Asymptotic behavior of mean uniform norms for sequences of Gaussian processes and fields

We consider the mean uniform (mixed) norms for a sequence of Gaussian random functions. For a wide class of Gaussian processes and fields the \(2\log n)^{1/2}\)-asymptotic for mixed norms is found whenever the volume of the index set is of order \(n\) and tends to infinity, for example, \(n\)-length time interval for random processes. Some numerical examples demonstrate the rate of convergence for the obtained asymptotic. The developed technique can be applied to analysis of various linear approximation methods. As an application we consider the rate of approximation by trigonometrical polynomials in the mean uniform norm.