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 a large class of non-centered smooth Gaussian random fields and provides a much more accurate approximation compared with those existing results by the double sum method. The explicit formulae for $\mathbb{E}\{χ(A_u(X,T))\}$ are also derived for two cases: (i) $T$ is a rectangle and $X-\mathbb{E} X$ is stationary; (ii) $T$ is an $N$-dimensional sphere and $X-\mathbb{E} X$ is isotropic.
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 $X= \{X(p), p\in M\}$ 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 …
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…
We derive a functional central limit theorem for the excursion of a random walk conditioned on sweeping a prescribed geometric area. We assume that the increments of the random walk are integer-valued, centered, with a third moment equal to zero and a finite fourth moment. This result complements the work of \citep{DKW13} where local central limit theorems are provided for the geometric area of the excursion of a symmetric random walk with finite second moments. Our result turns out to be a key tool to derive the scaling limit of the \emph{Interacting Partially-Directed Self-Avoiding Walk} at …
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…
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 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…
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…
In recent years, considerable interest has been drawn by the analysis of geometric functionals for the excursion sets of random eigenfunctions on the unit sphere (spherical harmonics). In this paper, we extend those results to proper subsets of the sphere $\mathbb{S}^2$, i.e., spherical caps, focussing in particular on the excursion area. Precisely, we show that the asymptotic behavior of the excursion area is dominated by the so-called second-order chaos component, and we exploit this result to establish a Quantitative Central Limit Theorem, in the high energy limit. These results generalize …
In a recent paper by Gorin and Shkolnikov (2016), they have found, as a corollary to their result relevant to random matrix theory, that the area below a normalized Brownian excursion minus one half of the integral of the square of its total local time, is identical in law with a centered Gaussian random variable with variance $1/12$. In this note, we give a pathwise interpretation to their identity; Jeulin's identity connecting normalized Brownian excursion and its local time plays an essential role in the exposition.
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.
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 …
We consider smooth, infinitely divisible random fields $(X(t),t\in M)$, $M\subset {\mathbb{R}}^d$, with regularly varying Levy measure, and are interested in the geometric characteristics of the excursion sets \[A_u=\{t\in M:X(t)>u\}\] over high levels u. For a large class of such random fields, we compute the $u\to\infty$ asymptotic joint distribution of the numbers of critical points, of various types, of X in $A_u$, conditional on $A_u$ being nonempty. This allows us, for example, to obtain the asymptotic conditional distribution of the Euler characteristic of the excursion set. In a signif…
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…
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…
In this paper we provide an upper bound for the conjunction probability of independent Gaussian smooth processes and then we prove that this bound is a good approximation with exponentially smaller error. Our result confirms the heuristic approximation by Euler characteristic method of Worsley and Friston and also implies the exact value of generalized Pickands constant in a special case. Some results for conjunction probability of correlated processes are also discussed.
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…
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…
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…
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.
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 study the exact constants in the moment inequalities for sums of centered independent random variables: improve their asymptotics, low and upper bounds, calculate more exact asymptotics, elaborate the numerical algorithm for their calculation, study the class of smoothing etc.
We construct obliquely reflected Brownian motions in all bounded simply connected planar domains, including non-smooth domains, with general reflection vector fields on the boundary. Conformal mappings and excursion theory are our main technical tools. A key intermediate step, which may be of independent interest, is an alternative characterization of reflected Brownian motions in smooth bounded planar domains with a given field of angles of oblique reflection on the boundary in terms of a pair of quantities, namely an integrable positive harmonic function, which represents the stationary dist…
Stein's method is used to approximate sums of discrete and locally dependent random variables by a centered and symmetric Binomial distribution. Under appropriate smoothness properties of the summands, the same order of accuracy as in the Berry-Essen Theorem is achieved. The approximation of the total number of points of a point processes is also considered. The results are applied to the exceedances of the $r$-scans process and to the Matérn hardcore point process type I.
In this paper we consider the distribution of the maximum of a Gaussian field defined on non locally convex sets. Adler and Taylor or Aza\"ıs and Wschebor give the expansions in the locally convex case. The present paper generalizes their results to the non locally convex case by giving a full expansion in dimension 2 and some generalizations in higher dimension. For a given class of sets, a Steiner formula is established and the correspondence between this formula and the tail of the maximum is proved. The main tool is a recent result of Aza\"ıs and Wschebor that shows that under some conditi…
In this paper we consider probabilistic analogues of some classical integral geometric formulae: Weyl--Steiner tube formulae and the Chern--Federer kinematic fundamental formula. The probabilistic building blocks are smooth, real-valued random fields built up from i.i.d. copies of centered, unit-variance smooth Gaussian fields on a manifold $M$. Specifically, we consider random fields of the form $f_p=F(y_1(p),...,y_k(p))$ for $F\in C^2(\mathbb{R}^k;\mathbb{R})$ and $(y_1,...,y_k)$ a vector of $C^2$ i.i.d. centered, unit-variance Gaussian fields. The analogue of the Weyl--Steiner formula for s…
Let $X(s,t), (s,t)\in E$, with $E\subset \mathbb{R}^2$ a compact set, be a centered two dimensional Gaussian random field with continuous trajectories and variance function $σ(s,t)$. Denote by $\mathcal{L}=\{(s,t): σ(s,t)=\max_{(s',t')\in E}σ(s',t')\}$. In this contribution, we derive the exact asymptotics of $\mathbb{P}\left(\sup_{(s,t)\in E}X(s,t)>u\right)$, as $u\to\infty$, under condition that $\mathcal{L}$ is a smooth curve. We illustrate our findings by an application concerned with extremes of the aggregation of two independent fractional Brownian motions.
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…
Let $\{X(t)= (X_1(t),X_2(t))^T,\ t \in \mathbb{R}^N\}$ be an $\mathbb{R}^2$-valued continuous locally stationary Gaussian random field with $\mathbb{E}[X(t)]=\mathbf{0}$. For any compact sets $A_1, A_2 \subset \mathbb{R}^N$, precise asymptotic behavior of the excursion probability \[ \mathbb{P}\bigg(\max_{s\in A_1} X_1(s)>u,\, \max_{t\in A_2} X_2(t)>u\bigg),\ \ \text{ as }\ u \rightarrow \infty \] is investigated by applying the double sum method. The explicit results depend not only on the smoothness parameters of the coordinate fields $X_1$ and $X_2$, but also on their maximum correlation $ρ…
We study the range $R_n$ of a random walk on the $d$-dimensional lattice $\mathbb{Z}^d$ indexed by a random tree with $n$ vertices. Under the assumption that the random walk is centered and has finite fourth moments, we prove in dimension $d\leq3$ that $n^{-d/4}R_n$ converges in distribution to the Lebesgue measure of the support of the integrated super-Brownian excursion (ISE). An auxiliary result shows that the suitably rescaled local times of the tree-indexed random walk converge in distribution to the density process of ISE. We obtain similar results for the range of critical branching ran…
Consider a doubly-infinite array of iid centered variables with moment conditions, from which one can extract a finite number of rectangular, overlapping submatrices, and form the corresponding Wishart matrices. We show that under basic smoothness assumptions, centered linear eigenstatistics of such matrices converge jointly to a Gaussian vector with an interesting covariance structure. This structure, which is similar to those appearing in work of Borodin, Borodin and Gorin, and Johnson and Pal can be described in terms of the height function, and leads to a connection with the Gaussian Free …
Let $ρ$ be compactly supported on $D \subset \mathbb R^2$. Endow $\mathbb R^2$ with the metric $e^ρ(dx_1^2 + dx_2^2)$. As $δ\to 0$ the set of Brownian loops centered in $D$ with length at least $δ$ has measure $$\frac{\text{area}(D)}{2πδ} + \frac{1}{48π}(ρ,ρ)_{\nabla}+ o(1).$$ When $ρ$ is smooth, this follows from the classical Polyakov-Alvarez formula. We show that the above also holds if $ρ$ is not smooth, e.g. if $ρ$ is only Lipschitz. This fact can alternatively be expressed in terms of heat kernel traces, eigenvalue asymptotics, or zeta regularized determinants. Variants of this statement…
This paper provides quantitative Central Limit Theorems for nonlinear transforms of spherical random fields, in the high frequency limit. The sequences of fields that we consider are represented as smoothed averages of spherical Gaussian eigenfunctions and can be viewed as random coefficients from continuous wavelets/needlets; as such, they are of immediate interest for spherical data analysis. In particular, we focus on so-called needlets polyspectra, which are popular tools for nonGaussianity analysis in the astrophysical community, and on the area of excursion sets. Our results are based on…
We compute the expected value of various quantities related to the biparametric singularities of a pair of smooth centered Gaussian random fields on an n-dimensional compact manifold, such as the lengths of the critical curves and contours of a fixed index and the number of cusps. We obtain certain expressions under no particular assumptions other than smoothness of the two fields, but more explicit formulae are derived under varying levels of additional constraints such as the two random fields being i.i.d, stationary, isotropic etc.
We are concerned with zeros of random power series with coefficients being a stationary, centered, complex Gaussian process. We show that the expected number of zeros in every smooth domain in the disk of convergence is less than that of the hyperbolic GAF with i.i.d. coefficients. When coefficients are finitely dependent, i.e., the spectral density is a trigonometric polynomial, we derive precise asymptotics of the expected number of zeros inside the disk of radius $r$ centered at the origin as $r$ tends to the radius of convergence, in the proof of which we clarify that the negative contribu…
The statistical description of the scalar conservation law of the form $ρ_t=H(ρ)_x$ with $H: \mathbb{R} \rightarrow \mathbb{R}$ a smooth convex function has been an object of interest when the initial profile $ρ(\cdot,0)$ is random. The special case when $H(ρ)=\frac{ρ^2}{2}$ (Burgers equation) has in particular received extensive interest in the past and is now understood for various random initial conditions. We solve in this paper a conjecture on the profile of the solution at any time $t>0$ for a general class of hamiltonians $H$ and show that it is a stationary piecewise-smooth Feller proc…
