Let $X= \{X(p), p\in M\}$ be a centered Gaussian random field, where $M$ is a smooth Riemannian manifold. For a suitable compact subset $D\subset M$, we obtain the approximations to excursion probability $\mathbb{P}\{\sup_{p\in D} X(p) \ge u \}$, as $u\to \infty$, for two cases: (i) $X$ is smooth and isotropic; (ii) $X$ is non-smooth and locally isotropic. For case (i), the expected Euler characteristic approximation is formulated explicitly; while for case (ii), it is shown that the asymptotics is similar to Pickands' approximation on Euclidean space which involves Pickands' constant and the volume of $D$. These extend the results in \citep{Cheng:2014} from sphere to general Riemannian manifolds.
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.
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…
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…
We study tail behaviour of the distribution of the area under the positive excursion of a random walk which has negative drift and heavy-tailed increments. We determine the asymptotics for tail probabilities for the area.
We develop a novel computational method for evaluating the extreme excursion probabilities arising for random initialization of nonlinear dynamical systems. The method uses a Markov chain Monte Carlo or a Laplace approximation approach to construct a biasing distribution that in turn is used in an importance sampling procedure to estimate the extreme excursion probabilities. The prior and likelihood of the biasing distribution are obtained by using Rice's formula from excursion probability theory. We use Gaussian mixture biasing distributions and approximate the non-Gaussian initial excitation…
We study tail behaviour of the distribution of the area under the positive excursion of a random walk which has negative drift and light-tailed increments. We determine the asymptotics for local probabilities for the area and prove a local central limit theorem for the duration of the excursion conditioned on the large values of its area.
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.
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…
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.
Our focus is on the design and analysis of efficient Monte Carlo methods for computing tail probabilities for the suprema of Gaussian random fields, along with conditional expectations of functionals of the fields given the existence of excursions above high levels, b. Naïve Monte Carlo takes an exponential, in b, computational cost to estimate these probabilities and conditional expectations for a prescribed relative accuracy. In contrast, our Monte Carlo procedures achieve, at worst, polynomial complexity in b, assuming only that the mean and covariance functions are Hölder continuous. We al…
We give an overview of the recent asymptotic results on the geometry of excursion sets of stationary random fields. Namely, we cover a number of limit theorems of central type for the volume of excursions of stationary (quasi--, positively or negatively) associated random fields with stochastically continuous realizations for a fixed excursion level. This class includes in particular Gaussian, Poisson shot noise, certain infinitely divisible, $α$--stable and max--stable random fields satisfying some extra dependence conditions. Functional limit theorems (with the excursion level being an argum…
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…
The structure of Gaussian random fields over high levels is a well researched and well understood area, particularly if the field is smooth. However, the question as to whether or not two or more points which lie in an excursion set belong to the same connected component has constantly eluded analysis. We study this problem from the point of view of large deviations, finding the asymptotic probabilities that two such points are connected by a path laying within the excursion set, and so belong to the same component. In addition, we obtain a characterization and descriptions of the most likely …
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…
We consider reflecting random walks on the nonnegative integers with drift of order 1/x at height x. We establish explicit asymptotics for various probabilities associated to such walks, including the distribution of the hitting time of 0 and first return time to 0, and the probability of being at a given height k at time n (uniformly in a large range of k.) In particular, for drift of form -δ/2x + o(1/x) with δ> -1, we show that the probability of a first return to 0 at time n is asymptotically n^{-c}ϕ(n), where c = (3+δ)/2 and ϕis a slowly varying function given explicitly in terms of the o(…
Let $X = \{X(t): t\in T \}$ be a non-centered, unit-variance, smooth Gaussian random field indexed on some parameter space $T$, and let $A_u(X,T) = \{t\in T: X(t)\geq u\}$ be the excursion set of $X$ exceeding level $u$. Under certain smoothness and regularity conditions, it is shown that, as $u\to \infty$, the excursion probability $\mathbb{P}\{\sup_{t\in T} X(t)\ge u \}$ can be approximated by the expected Euler characteristic of $A_u(X,T)$, denoted by $\mathbb{E}\{χ(A_u(X,T))\}$, such that the error is super-exponentially small. This verifies the expected Euler characteristic heuristic for …
In this paper, we shall be concerned with geometric functionals and excursion probabilities for some nonlinear transforms evaluated on Fourier components of spherical random fields. In particular, we consider both random spherical harmonics and their smoothed averages, which can be viewed as random wavelet coefficients in the continuous case. For such fields, we consider smoothed polynomial transforms; we focus on the geometry of their excursion sets, and we study their asymptotic behaviour, in the high-frequency sense. We focus on the analysis of Euler-Poincaré characteristics, which can be e…
In this paper, we consider isotropic and stationary real Gaussian random fields defined on $\mathbb{S}^2\times\mathbb{R}$ and we investigate the asymptotic behavior, as $T\rightarrow +\infty$, of the empirical measure (excursion area) in $\mathbb{S}^2\times\mathbb{R}$ at any threshold, covering both cases when the field exhibits short and long memory, i.e. integrable and non-integrable temporal covariance. It turns out that the limiting distribution is not universal, depending both on the memory parameters and the threshold. In particular, in the long memory case a form of Berry's cancellation…
In this paper, we consider a classic problem concerning the high excursion probabilities of a Gaussian random field $f$ living on a compact set $T$. We develop efficient computational methods for the tail probabilities $P(\sup_T f(t) > b)$ and the conditional expectations $E(Γ(f) | \sup_T f(t) > b)$ as $b\rightarrow \infty$. For each $\varepsilon$ positive, we present Monte Carlo algorithms that run in \emph{constant} time and compute the interesting quantities with $\varepsilon$ relative error for arbitrarily large $b$. The efficiency results are applicable to a large class of Hölder continuo…
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 determine the large exceedance probabilities and large exceedance paths for the matrix recursive sequence $V_n = M_n V_{n-1} + Q_n, \: n=1,2,\ldots,$ where $\{M_n\}$ is an i.i.d. sequence of $d \times d$ random matrices and $\{ Q_n\}$ is an i.i.d. sequence of random vectors, both with nonnegative entries. Early work on this problem dates to Kesten's (1973) seminal paper, motivated by an application to multi-type branching processes. Other applications arise in financial time series modeling (connected to the study of the GARCH($p,q$) processes) and in physics, and this recursive sequence ha…
We consider one-dimensional random walks in random environment which are transient to the right. Our main interest is in the study of the sub-ballistic regime, where at time $n$ the particle is typically at a distance of order $O(n^κ)$ from the origin, $κ\in(0,1)$. We investigate the probabilities of moderate deviations from this behaviour. Specifically, we are interested in quenched and annealed probabilities of slowdown (at time $n$, the particle is at a distance of order $O(n^{ν_0})$ from the origin, $ν_0\in (0,κ)$), and speedup (at time $n$, the particle is at a distance of order $n^{ν_1}$…
In the first part of this article, we prove two-sided estimates of hitting probabilities of balls, the potential kernel and the Green function for a ball for general isotropic unimodal Lévy processes. Our bounds are sharp under the absence of the Gaussian component and a mild regularity condition on the density of the Lévy measure: its radial profile needs to satisfy a scaling-type condition, which is equivalent to $O$-regular variation at zero and at infinity with lower indices greater than $-d - 2$. We also prove a supremum estimate and a regularity result for functions harmonic with respect…
We study exit laws from large balls in $\mathbb{Z}^d$, $d\geq3$, of random walks in an i.i.d. random environment that is a small perturbation of the environment corresponding to simple random walk. Under a centering condition on the measure governing the environment, we prove that the exit laws are close to those of a symmetric random walk, which we identify as a perturbed simple random walk. We obtain bounds on total variation distances as well as local results comparing exit probabilities on boundary segments. As an application, we prove transience of the random walks in random environment. …
In this article, we study the excursions sets $\mathcal{D}\_p=f^{-1}([-p,+\infty[)$ where $f$ is a natural real-analytic planar Gaussian field called the Bargmann-Fock field. More precisely, $f$ is the centered Gaussian field on $\mathbb{R}^2$ with covariance $(x,y) \mapsto \exp(-\frac{1}{2}|x-y|^2)$. In [BG16], Beffara and Gayet prove that, if $p \leq 0$, then a.s. $\mathcal{D}\_p$ has no unbounded component. We show that conversely, if $p>0$, then a.s. $\mathcal{D}\_p$ has a unique unbounded component. As a result, the critical level of this percolation model is $0$. We also prove exponentia…
We provide a generalization of the Gaussian Kinematic Formula (GKF) in Taylor(2006) for multivariate, heterogeneous Gaussian-related fields. The fields under consideration are non-Gaussian fields built out of smooth, independent Gaussian fields with heterogeneity in distribution amongst the individual building blocks. Our motivation comes from potential applications in the analysis of Cosmological Data (CMB). Specifically, future CMB experiments will be focusing on polarization data, typically modeled as isotropic vector-valued Gaussian related fields with independent, but non-identically dist…
Let f be a stationary isotropic non-degenerate Gaussian field on R^2. Assume that f = q * W where q is both C^2 and L^2 and W is the L^2 white noise on R^2. We extend a result by Stephen Muirhead and Hugo Vanneuville by showing that, assuming that q * q is pointwise non-negative and has fast enough decay, the set {f > -l} percolates with probability one when l > 0 and with probability zero if l < 0 or l = 0. We also prove exponential decay of crossing probabilities and uniqueness of the unbounded cluster. To this end, we study a Gaussian field g defined on the torus and establish a superconcen…
Let $\{f(t): t\in T\}$ be a smooth Gaussian random field over a parameter space $T$, where $T$ may be a subset of Euclidean space or, more generally, a Riemannian manifold. For any local maximum of $f(t)$ located at $t_0$ in the interior of $T$, we provide general formulae and asymptotic approximations for both the tail distribution of the height of a local maximum $\mathbb{P}\{f(t_0)>u | t_0 \text{is a local maximum of} f(t) \}$ and the overshoot distribution of a local maximum $\mathbb{P}\{f(t_0)>u+v | t_0 \text{is a local maximum of} f(t) \text{and} f(t_0)>v\}$. Assuming further that $f$ is…
This paper deals with the problem of detecting non-isotropic high-dimensional geometric structure in random graphs. Namely, we study a model of a random geometric graph in which vertices correspond to points generated randomly and independently from a non-isotropic $d$-dimensional Gaussian distribution, and two vertices are connected if the distance between them is smaller than some pre-specified threshold. We derive new notions of dimensionality which depend upon the eigenvalues of the covariance of the Gaussian distribution. If $α$ denotes the vector of eigenvalues, and $n$ is the number of …
The concept of typical and weighted typical spherical faces for tessellations of the $d$-dimensional unit sphere, generated by $n$ independent random great hyperspheres distributed according to a non-degenerate directional distribution, is introduced and studied. Probabilistic interpretations for such spherical faces are given and their directional distributions are determined. Explicit formulas for the expected $f$-vector, the expected spherical Quermaßintegrals and the expected spherical intrinsic volumes are found in the isotropic case. Their limiting behaviour as $n\to\infty$ is discussed …
