This is an extended version of a series of talks I held at the University of Bochum in 2017 about limit theorems for non-linear functionals of stationary Gaussian random fields.
The goal of these talks was to give a fairly detailed introduction to the theory leading to such results, even if some of the results are presented without proof. On the other hand, I gave a simpler proof for some of the results. (The proofs omitted from this text can be found in my Springer Lecture Note Multiple Wiener--Ito Integrals.
In this note first I discuss the spectral representation of the covariance function of a Gaussian stationary rendom field by means of the spectral measure and the representation of the elements of the random field by means of a random integral with respect to the random spectral measure. Then I construct the multiple random integrals with respect to the random spectral measure and prove their most important properties. Finally I show some interesting applications of these multiple random integrals. In particular, I prove some non-trivial non-Gaussian limit theorems
We deduce bilateral interrelations between Fourier coefficients for lacunar trigonometric series and modulus of their continuity. We obtain also as an application some conditions for continuity and discontinuity for Gaussian periodic stationary random centered processes.
With motivation from K. Dȩbicki and P. Kisowski (2007), in this paper we derive the exact tail asymptotics of $α(t)$-locally stationary Gaussian processes with non-constant variance functions. We show that some certain variance functions lead to qualitatively new results.
This paper studies the problem of equivalence of Gaussian measures induced by Gaussian random fields (GRFs) with stationary increments and proves a sufficient condition for the equivalence in terms of the behavior of the spectral measures at infinity. The main results extend those of Stein (2004), Van Zanten (2007, 2008) and are applicable to a rich family of nonstationary space-time models with possible anisotropy behavior.
Depending on a parameter $h\in (0,1]$, let $\{X_h(\mathbf{t})$, $\mathbf{t}\in\mathcal{M}_h\}$ be a class of centered Gaussian fields indexed by compact manifolds $\mathcal{M}_h$. For locally stationary Gaussian fields $X_h$, we study the asymptotic excursion probabilities of $X_h$ on $\mathcal{M}_h$. Two cases are considered: (i) $h$ is fixed and (ii) $h\rightarrow0$. These results are extended to obtain the limit behaviors of the extremes of locally stationary $χ$-fields on manifolds.
We describe two classes of Gaussian self-similar random fields: with strictly stationary rectangular increments and with mild stationary rectangular increments. We find explicit spectral and moving average representations for the fields with strictly stationary rectangular increments and characterize fields with mild stationary rectangular increments by the properties of covariance functions of their Lamperti transformations as well as in terms of their spectral densities. We establish that both classes contain not only fractional Brownian sheets and we provide corresponding examples. As a by-…
Assuming certain conditions on the spectral measures of centered stationary Gaussian processes on $\mathbb{R}$ (or ${\mathbb{R}}^2$), we show that the probability of the event that their zero count in an interval (resp., nodal length in a square domain) is larger than $n$, where $n$ is much larger than the expected value of the zero count in that interval (resp., nodal length in that square domain), is exponentially small in $n$.
We study the persistence probability for some two-sided discrete-time Gaussian sequences that are discrete-time analogs of fractional Brownian motion and integrated fractional Brownian motion, respectively. Our results extend the corresponding ones in continuous-time in [Molchan, Commun. Math. Phys., 1999] and [Molchen, J. Stat. Phys., 2017] to a wide class of discrete-time processes.
We study the Langevin equation with stationary-increment Gaussian noise. We show the strong consistency and the asymptotic normality with Berry--Esseen bound of the so-called alternative estimator of the mean reversion parameter. The conditions and results are stated in terms of the variance function of the noise. We consider both the case of continuous and discrete observations. As examples we consider fractional and bifractional Ornstein--Uhlenbeck processes. Finally, we discuss the maximum likelihood and the least squares estimators.
In this paper we study the local times of vector-valued Gaussian fields that are `diagonally operator-self-similar' and whose increments are stationary. Denoting the local time of such a Gaussian field around the spatial origin and over the temporal unit hypercube by $Z$, we show that there exists $λ\in(0,1)$ such that under some quite weak conditions, $\lim_{n\rightarrow +\infty}\frac{\sqrt[n]{\mathbb{E}(Z^n)}}{n^λ}$ and $\lim_{x\rightarrow +\infty}\frac{-\log \mathbb{P}(Z>x)}{x^{\frac{1}λ}}$ both exist and are strictly positive (possibly $+\infty$). Moreover, we show that if the underlying G…
Given that a stationary Gaussian process is above a high threshold, the length of time it spends before going below that threshold is studied. The asymptotic order is determined by the smoothness of the sample paths, which in turn is a function of the tails of the spectral measure. Two disjoint regimes are studied - one in which the second spectral moment is finite and the other in which the tails of the spectral measure are regularly varying and the second moment is infinite.
Lower bounds for persistence probabilities of stationary Gaussian processes in discrete time are obtained under various conditions on the spectral measure of the process. Examples are given to show that the persistence probability can decay faster than exponentially. It is shown that if the spectral measure is not singular, then the exponent in the persistence probability cannot grow faster than quadratically. An example that appears (from numerical evidence) to achieve this lower bound is presented.
We prove convergence of Hausdorff measure of level sets of smooth Gaussian fields when the levels converge. Given two coupled stationary fields $f_1, f_2$ , we estimate the difference of Hausdorff measure of level sets in expectation, in terms of $C^2$-fluctuations of the field $F=f_1-f_2$. The main idea in the proof is to represent difference in volume as an integral of mean curvature using the divergence theorem. This approach is different from using Kac-Rice type formula as main tool in the analysis.
We consider the small deviation probabilities (SDP) for sums of stationary Gaussian sequences. For the cases of constant boundaries and boundaries tending to zero, we obtain quite general results. For the case of the boundaries tending to infinity, we focus our attention on the discrete analogs of the fractional Brownian motion (FBM). It turns out that the lower bounds for the SDP can be transferred from the well studied FBM caseto the discrete time setting under the usual assumptions that imply weak convergence while the transfer of the corresponding upper bounds necessarily requires a deeper…
Let $\{X_i(t):\, t\in S\subset \R^d \}_{i=1,2,\ldots,n}$ be independent copies of a stationary centered Gaussian field with almost surely smooth sample paths. In this paper, we are interested in the conjunction probability defined as $\PP \left( \exists t\in S: \, X_i(t) \geq u, \, \forall i=1,2,\ldots,n \right)$ for a given threshold level $u$. As $u\rightarrow \infty$, we will provide an asymptotic formula for the conjunction probability. This asymptotic formula is derived from the behaviour of the volume of the set of local maximum points. The proof relies on a result of Aza\"ıs and Wschebo…
Persistence and entropic repulsion of stationary Gaussian fields with spectral singularity at the origin
Bar-Ilan University（巴伊兰大学）；Einstein Institute of Mathematics, Hebrew University of Jerusalem（耶路撒冷希伯来大学爱因斯坦数学研究所）；School of Mathematics, Monash University（蒙纳士大学数学学院）
The paper deals with the regression model $X_t = θt + B_t$, $t\in[0, T ]$, where $B=\{B_t, t\geq 0\}$ is a centered Gaussian process with stationary increments. We study the estimation of the unknown parameter $θ$ and establish the formula for the likelihood function in terms of a solution to an integral equation. Then we find the maximum likelihood estimator and prove its strong consistency. The results obtained generalize the known results for fractional and mixed fractional Brownian motion.
We investigate asymptotics of the tail distribution of sojourn time $$ \int_0^T \mathbb{I}(X(t)> u)dt, $$ as $u\to\infty$, where $X$ is a centered stationary Gaussian process and $T$ is an independent of $X$ nonnegative random variable. The heaviness of the tail distribution of $T$ impacts the form of the asymptotics, leading to four scenarios: the case of integrable $T$, the case of regularly varying $T$ with index $λ=1$ and index $λ\in(0,1)$ and the case of slowly varying tail distribution of $T$. The derived findings are illustrated by the analysis of the class of fractional Ornstein-Uhlenb…
Let $\{Z(τ,s), (τ,s)\in [a,b]\times[0,T]\}$ with some positive constants $a,b,T$ be a centered Gaussian random field with variance function $σ^{2}(τ,s)$ satisfying $σ^{2}(τ,s)=σ^{2}(τ)$. We firstly derive the exact tail asymptotics for the maximum $M_{H}(T)=\max_{(τ,s)\in[a,b]\times[0,T]}Z(τ,s)/σ(τ)$ up crossing some level $u$ with any fixed $0<a<b<\infty$ and $T>0$; and we further derive the extreme limit law for $M_{H}(T)$. As applications of the main results, we derive the exact tail asymptotics and the extreme limit law for Shepp statistics with stationary Gaussian process, fractional Brow…
We prove a functional central limit theorem for integrals $\int_W f(X(t))\, dt$, where $(X(t))_{t\in\mathbb{R}^d}$ is a stationary mixing random field and the stochastic process is indexed by the function $f$, as the integration domain $W$ grows in Van Hove-sense. We discuss properties of the covariance function of the asymptotic Gaussian process.
We consider the winding number of planar stationary Gaussian processes defined on the line. Under mild conditions, we obtain the asymptotic variance and the Central Limit Theorem for the winding number as the time horizon tends to infinity. In the asymptotic regime, our discrete approach is equivalent to the continuous one studied previously in the literature and our main result extends the existing ones. Our model allows for a general dependence of the coordinates of the process and non-differentiability of one of them. Furthermore, beyond our general framework, we consider as examples an app…
In this paper we obtain new limit theorems for variational functionals of high frequency observations of stationary increments Lévy driven moving averages. We will see that the asymptotic behaviour of such functionals heavily depends on the kernel, the driving Lévy process and the properties of the functional under consideration. We show the "law of large numbers" for our class of statistics, which consists of three different cases. For one of the appearing limits, which we refer to as the ergodic type limit, we also prove the associated weak limit theory, which again consists of three differe…
We consider a class of non-homogeneous, continuous, centered Gaussian random fields $\{X_h(t), t \in {\cal M}_h;\,0 < h \le 1\}$ where ${\cal M}_h$ denotes a rescaled smooth manifold, i.e. ${\cal M}_h = \frac{1}{h} {\cal M},$ and study the limit behavior of the extreme values of these Gaussian random fields when $h$ tends to zero, which means that the manifold is growing. Our main result can be thought of as a generalization of a classical result of Bickel and Rosenblatt (1973a), and also of results by Mikhaleva and Piterbarg (1997).
This is a survey of recent results on central and non-central limit theorems for quadratic functionals of stationary processes. The underlying processes are Gaussian, linear or Lévy-driven linear processes with memory, and are defined either in discrete or continuous time. We focus on limit theorems for Toeplitz and tapered Toeplitz type quadratic functionals of stationary processes with applications in parametric and nonparametric statistical estimation theory. We discuss questions concerning Toeplitz matrices and operators, Fejér-type singular integrals, and Lévy-Itô-type and Stratonovich-ty…
The seminal papers of Pickands [1,2] paved the way for a systematic study of high exceedance probabilities of both stationary and non-stationary Gaussian processes. Yet, in the vector-valued setting, due to the lack of key tools including Slepian's Lemma, Borell-TIS and Piterbarg inequalities there has not been any methodological development in the literature for the study of extremes of vector-valued Gaussian processes. In this contribution we develop the uniform double-sum method for the vector-valued setting obtaining the exact asymptotics of the exceedance probabilities for both stationary…
In this paper, we consider the distribution of the supremum of non-stationary Gaussian processes, and present a new theoretical result on the asymptotic behaviour of this distribution. Unlike previously known facts in this field, our main theorem yields the asymptotic representation of the corresponding distribution function with exponentially decaying remainder term. This result can be efficiently used for studying the projection density estimates, based, for instance, on Legendre polynomials. More precisely, we construct the sequence of accompanying laws, which approximates the distribution …
For a given centered Gaussian process with stationary increments $\{X(t), t\geq 0\}$ and $c>0$, let $$ W_γ(t)=X(t)-ct-γ\inf_{0\leq s\leq t}\left(X(s)-cs\right), \quad t\geq 0$$ denote the $γ$-reflected process, where $γ\in (0,1)$. This process is introduced in the context of risk theory to model surplus process that include tax payments of loss-carry forward type.In this contribution we derive asymptotic approximations of both the ruin probability and the joint distribution of first and last passage times given that ruin occurs. We apply our findings to the cases with $X$ being the multiplex f…
The purpose of this note is to show that it is possible to establish a many-server functional strong law of large numbers (FSLLN) for the fraction of occupied servers (i.e., the scaled number-in-system) without explicitly tracking either the age or the residual service times of the jobs in a non-Markovian, non-stationary loss model. This considerable analytical simplification is achieved by exploiting a semimartingale representation. The fluid limit is shown to be the unique solution of a Volterra integral equation.
