This paper considers the asymptotic behaviour of volumes of excursion sets of subordinated Gaussian random fields with (possibly) infinite variance. Actually, we consider integral functionals of such fields and obtain their limiting distribution using the Hermite expansion of the integrand. We consider the general non-stationary Gaussian random fields, including stationary and anisotropic special cases. The limiting random variables in our limit theorems have the form of multiple Wiener-Itô integrals. We illustrate most results with corresponding examples.
We compute quantitative bounds for measuring the discrepancy between the distribution of two min-max statistics involving either pairs of Gaussian random matrices, or one Gaussian and one Gaussian-subordinated random matrix. In the fully Gaussian setup, our approach allows us to recover quantitative versions of well-known inequalities by Gordon (1985, 1987, 1992), thus generalising the quantitative version of the Sudakov-Fernique inequality deduced in Chatterjee (2005). On the other hand, the Gaussian-subordinated case yields generalizations of estimates by Chernozhukov et al. (2015) and Koike…
Motivated by the subordinated Brownian motion, we define a new class of (in general discontinuous) random fields on higher-dimensional parameter domains: the subordinated Gaussian random field. We investigate the pointwise marginal distribution of the constructed random fields, derive a Lévy-Khinchin-type formula and semi-explicit formulas for the covariance function. Further, we study the pointwise stochastic regularity and validate our theoretical findings in various numerical examples.
We consider Gaussian subordinated Lévy fields (GSLFs) that arise by subordinating Lévy processes with positive transformations of Gaussian random fields on some spatial domain $\mathcal{D}\subset \mathbb{R}^d$, $d\geq 1$. The resulting random fields are distributionally flexible and have in general discontinuous sample paths. Theoretical investigations of the random fields include pointwise distributions, possible approximations and their covariance function. As an application, a random elliptic PDE is considered, where the constructed random fields occur in the diffusion coefficient. Further,…
Our interest in this paper is to explore limit theorems for various geometric functionals of excursion sets of isotropic Gaussian random fields. In the past, limit theorems have been proven for various geometric functionals of excursion sets/sojourn times. Most recently a CLT for Euler-Poincaré characteristic of the excursions set of a Gaussian random field has been proven under appropriate conditions. In this paper, we shall obtain a central limit theorem for some global geometric functionals, called the Lipschitz-Killing curvatures of excursion sets of Gaussian random fields in an appropriat…
Gaussian random fields on finite dimensional smooth manifolds whose variances reach their maximum value at smooth submanifolds are considered. Exact asymptotic behaviors of large excursion probabilities have been evaluated. Vector Gaussian processes, chi-square processes, Bessel process, fractional Bessel process, Bessel bridge are examples of application of this result.
In this article, we show that a general class of weakly stationary time series can be modeled applying Gaussian subordinated processes. We show that, for any given weakly stationary time series $(z_t)_{z\in\mathbb{N}}$ with given equal one-dimensional marginal distribution, one can always construct a function $f$ and a Gaussian process $(X_t)_{t\in\mathbb{N}}$ such that $\left(f(X_t)\right)_{t\in\mathbb{N}}$ has the same marginal distributions and, asymptotically, the same autocovariance function as $(z_t)_{t\in\mathbb{N}}$. Consequently, we obtain asymptotic distributions for the mean and aut…
This paper studies the excursion set of a real stationary isotropic Gaussian random field above a fixed level. We show that the standardized Lipschitz-Killing curvatures of the intersection of the excursion set with a window converges in distribution to a normal distribution as the window grows to the $d$-dimensional Euclidean space. Moreover a lower bound for the asymptotic variance is shown.
Let $\{(X(t), Y(s)): t\in T, s\in S\}$ be an $\mathbb{R}^2$-valued, centered, unit-variance smooth Gaussian vector field, where $T$ and $S$ are compact rectangles in $\mathbb{R}^N$. It is shown that, as $u\to \infty$, the joint excursion probability $\mathbb{P} \{\sup_{t\in T} X(t) \geq u, \sup_{s\in S} Y(s) \geq u \}$ can be approximated by $\mathbb{E}\{χ(A_u)\}$, the expected Euler characteristic of the excursion set $A_u=\{(t,s)\in T\times S: X(t) \ge u, Y(s) \ge u\}$, such that the error is super-exponentially small. This verifies the expected Euler characteristic heuristic (cf. Taylor, Ta…
Random field excursions is an increasingly vital topic within data analysis in medicine, cosmology, materials science, etc. This work is the first detailed study of their Betti numbers in the so-called `sparse' regime. Specifically, we consider a piecewise constant Gaussian field whose covariance function is positive and satisfies some local, boundedness, and decay rate conditions. We model its excursion set via a Cech complex. For Betti numbers of this complex, we then prove various limit theorems as the window size and the excursion level together grow to infinity. Our results include asympt…
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.
We investigate the asymptotic variance of Gaussian nodal excursions in the Euclidean space, focusing on the case where the spectral measure has incommensurable atoms. This study requires to establish fine recurrence properties in 0 for the associated irrational random walk on the torus. We show in particular that the recurrence magnitude depends strongly on the diophantine properties of the atoms, and the same goes for the variance asymptotics of nodal excursions. More specifically, if the spectral measures contains atoms which ratios are well approximable by rationals, the variance is likely …
For a smooth, stationary, planar Gaussian field, we consider the number of connected components of its excursion set (or level set) contained in a large square of area $R^2$. The mean number of components is known to be of order $R^2$ for generic fields and all levels. We show that for certain fields with positive spectral density near the origin (including the Bargmann-Fock field), and for certain levels $\ell$, these random variables have fluctuations of order at least $R$, and hence variance of order at least $R^2$. In particular, this holds for excursion sets when $\ell$ is in some neighbo…
In this paper, we investigate some geometric functionals for band limited Gaussian and isotropic spherical random fields in dimension 2. In particular, we focus on the area of excursion sets, providing its behavior in the high energy limit. Our result is based on Wiener chaos expansion for non linear transform of Gaussian fields and on an explicit derivation on the high-frequency limit of the covariance function of the field. As a simple corollary we establish also the Central Limit Theorem for the excursion area.
Let M be a compact smooth manifold of dimension n with or without boundary, and f : M $\rightarrow$ R be a smooth Gaussian random field. It is very natural to suppose that for a large positive real u, the random excursion set {f $\ge$ u} is mostly composed of a union of disjoint topological n-balls. Using the constructive part of (stratified) Morse theory we prove that in average, this intuition is true, and provide for large u the asymptotic of the expected number of such balls, and so of connected components of {f $\ge$ u}, see Theorem 1.2. We similarly show that in average, the high nodal s…
The Nazarov-Sodin constant describes the average number of nodal set components of Gaussian fields on large scales. We generalise this to a functional describing the corresponding number of level set components for arbitrary levels. Using results from Morse theory, we express this functional as an integral over the level densities of different types of critical points, and as a result deduce the absolute continuity of the functional as the level varies. We further give upper and lower bounds showing that the functional is at least bimodal for certain isotropic fields, including the important s…
Nazarov and Sodin have shown that the number of connected components of the nodal set of a planar Gaussian field in a ball of radius $R$, normalised by area, converges to a constant as $R\to \infty $. This has been generalised to excursion/level sets at arbitrary levels, implying the existence of functionals $c_{ES}(\ell )$ and $c_{LS}(\ell )$ that encode the density of excursion/level set components at the level $\ell $. We prove that these functionals are continuously differentiable for a wide class of fields. This follows from a more general result, which derives differentiability of the fu…
To model subsurface flow in uncertain heterogeneous\ fractured media an elliptic equation with a discontinuous stochastic diffusion coefficient - also called random field - may be used. In case of a one-dimensional parameter space, Lévy processes allow for jumps and display great flexibility in the distributions used. However, in various situations (e.g. microstructure modeling), a one-dimensional parameter space is not sufficient. Classical extensions of Lévy processes on two parameter dimensions suffer from the fact that they do not allow for spatial discontinuities. In this paper a new subo…
Let $X=\{X(x): x\in\mathbb{S}^N\}$ be a real-valued, centered Gaussian random field indexed on the $N$-dimensional unit sphere $\mathbb{S}^N$. Approximations to the excursion probability ${\mathbb{P}}\{\sup_{x\in\mathbb{S}^N}X(x)\ge u\}$, as $u\to\infty$, are obtained for two cases: (i) $X$ is locally isotropic and its sample functions are non-smooth and; (ii) $X$ is isotropic and its sample functions are twice differentiable. For case (i), the excursion probability can be studied by applying the results in Piterbarg (Asymptotic Methods in the Theory of Gaussian Processes and Fields (1996) Ame…
We prove the existence of phase transitions in the global connectivity of the excursion sets of planar symmetric shot noise fields. Our main result establishes a phase transition with respect to the level for shot noise fields with symmetric log-concave mark distributions, including Gaussian, uniform, and Laplace marks, and kernels that are positive, symmetric, and have sufficient tail decay. Without the log-concavity assumption we prove a phase transition with respect to the intensity of positive marks.
We investigate Lipschitz-Killing curvatures for excursion sets of random fields on $\mathbb R^2$ under small spatial-invariant random perturbations. An expansion formula for mean curvatures is derived when the magnitude of the perturbation vanishes, which recovers the Gaussian Kinematic Formula at the limit by contiguity of the model. We develop an asymptotic study of the perturbed excursion area behaviour that leads to a quantitative non-Gaussian limit theorem, in Wasserstein distance, for fixed small perturbations and growing domain. When letting both the perturbation vanish and the domain g…
This work is to popularize the method of computing the distribution of the excursion times for a Gaussian process that involves extended and multivariate Rice's formula. The approach was used in numerical implementations of the high-dimensional integration routine and in earlier work it was shown that the computations are more effective and thus more precise than those based on Rice expansions. The joint distribution of excursion times is related to the distribution of the number of level crossings, a problem that can be attacked via the Rice series expansion, based on the moments of the numbe…
A deterministic application $θ\,:\,\mathbb{R}^2\rightarrow\mathbb{R}^2$ deforms bijectively and regularly the plane and allows to build a deformed random field $X\circθ\,:\,\mathbb{R}^2\rightarrow\mathbb{R}$ from a regular, stationary and isotropic random field $X\,:\,\mathbb{R}^2\rightarrow\mathbb{R}$. The deformed field $X\circθ$ is in general not isotropic, however we give an explicit characterization of the deformations $θ$ that preserve the isotropy. Further assuming that $X$ is Gaussian, we introduce a weak form of isotropy of the field $X\circθ$, defined by an invariance property of the…
In the last two decades there was a lot of progress in understanding the geometry of smooth Gaussian fields. This survey aims to cover one particular line of research: the large scale behaviour of level and excursion sets and their (conjectured) connection to the percolation theory.
We consider the Gaussian free field $φ$ on $\mathbb{Z}^d$, for $d\geq3$, and give sharp bounds on the probability that the radius of a finite cluster in the excursion set $\{φ\geq h\}$ exceeds a large value $N$, for any height $h \neq h_*$, where $h_*$ refers to the corresponding percolation critical parameter. In dimension $d=3$, we prove that this probability is sub-exponential in $N$ and decays as $\exp\{-\fracπ{6}(h-h_*)^2 \frac{N}{\log N} \}$ as $N \to \infty$ to principal exponential order. When $d\geq 4$, we prove that these tails decay exponentially in $N$. Our results extend to other …
Let $X=\{X(t),t\in {\mathbb{R}}^N\}$ be a centered Gaussian random field with stationary increments and $X(0)=0$. For any compact rectangle $T\subset {\mathbb{R}}^N$ and $u\in {\mathbb{R}}$, denote by $A_u=\{t\in T:X(t)\geq u\}$ the excursion set. Under $X(\cdot)\in C^2({\mathbb{R}}^N)$ and certain regularity conditions, the mean Euler characteristic of $A_u$, denoted by ${\mathbb{E}}\{φ(A_u)\}$, is derived. By applying the Rice method, it is shown that, as $u\to\infty$, the excursion probability ${\mathbb{P}}\{\sup_{t\in T}X(t)\geq u\}$ can be approximated by ${\mathbb{E}}\{φ(A_u)\}$ such tha…
