Asymptotic formula for the conjunction probability of smooth stationary Gaussian fields

Stationary IPA Estimates for Non-Smooth G/G/1/$\infty$ Functionals via Palm Inversion and Level-Crossing Analysis

We give stationary estimates for the derivative of the expectation of a non-smooth function of bounded variation f of the workload in a G/G/1/$\infty$ queue, with respect to a parameter influencing the distribu- tion of the input process. For this, we use an idea of Konstantopoulos and Zazanis based on the Palm inversion formula, however avoiding a limiting argument by performing the level-crossing analysis thereof globally, via Fubini's theorem. This method of proof allows to treat the case where the workload distribution has a mass at discontinuities of f and where the formula has to be modi…

Extremes of Order Statistics of Stationary Processes

Let $\{X_i(t),t\ge0\}, 1\le i\le n$ be independent copies of a stationary process $\{X(t), t\ge0\}$. For given positive constants $u,T$, define the set of $r$th conjunctions $ C_{r,T,u}:= \{t\in [0,T]: X_{r:n}(t) > u\}$ with $X_{r:n}(t)$ the $r$th largest order statistics of $X_1(t), \ldots , X_n(t), t\ge 0$. In numerous applications such as brain mapping and digital communication systems, of interest is the approximation of the probability that the set of conjunctions $C_{r,T,u}$ is not empty. Imposing the Albin's conditions on $X$, in this paper we obtain an exact asymptotic expansion of thi…

关于光滑高斯非平稳过程与带趋势的平稳平面场的局部极大值

On local maxima of smooth Gaussian nonstationary processes and stationary planar fields with trends

本文研究非中心光滑高斯场中局部极大值的期望数量与高度分布，针对非平稳过程引入条件相关参数简化计算，并通过广义卡方密度函数处理带趋势的非各向同性平稳平面场，为信号与变点检测提供p值和功效计算工具。

On a stationary random knot

In this article the construction of a stationary random knot is proposed. The corresponding smooth random curve has no self-intersections in deterministic moments of time and changes its topological type at random moments.

Large deviations for stationary measures of stochastic nonlinear wave equation with smooth white noise

We prove the Freidlin-Wentzell type large deviations principle for the family of stationary measures of stochastic nonlinear wave (NLW) equation with white noise. We do not assume that the limiting equation possesses a unique equilibrium and do not impose roughness on the noise. This allows to provide the first such result in the PDE setting.

Asymptotic formula for the tail of the maximum of smooth Stationary Gaussian fields on non locally convex sets

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…

平稳光滑高斯场的高局部极大值

High local maxima of stationary smooth Gaussian fields

研究平稳光滑高斯场在高阈值下的局部极大值分布，证明其弱收敛于泊松过程并给出量化中心极限定理。

多维平稳时间序列具有光滑特征值的谱密度的因式分解

Factorization of a spectral density with smooth eigenvalues of a multidimensional stationary time series

本文将一维光滑谱密度的经典结果推广到多维情形，通过显式因式分解谱密度来计算Wold表示的组成部分，从而为多维平稳时间序列的最佳线性预测提供理论基础。

Zeros of smooth stationary Gaussian processes

Let $f:\mathbb{R} \to \mathbb{R}$ be a stationary centered Gaussian process. For any $R>0$, let $ν_R$ denote the counting measure of $\{x \in \mathbb{R} \mid f(Rx)=0\}$. In this paper, we study the large $R$ asymptotic distribution of $ν_R$. Under suitable assumptions on the regularity of $f$ and the decay of its correlation function at infinity, we derive the asymptotics as $R \to +\infty$ of the central moments of the linear statistics of $ν_R$. In particular, we derive an asymptotics of order $R^\frac{p}{2}$ for the $p$-th central moment of the number of zeros of $f$ in $[0,R]$. As an appli…

关于分数噪声驱动随机微分方程的（非）平稳密度

On the (Non-)Stationary Density of Fractional-Driven Stochastic Differential Equations

本文利用 Wiener-Liouville 桥表示证明任意 Hurst 参数加性分数噪声 SDE 的平稳与非平稳密度具有 Gaussian 界及参数联合光滑性，并移除分数平均原理中的特设假设。

Hölder regularity of stationary measures

We consider smooth random dynamical systems defined by a distribution with a finite moment of the norm of the differential, and prove that under suitable non-degeneracy conditions any stationary measure must be Hölder continuous. The result is a vast generalization of the classical statement on Hölder continuity of stationary measures of random walks on linear groups.

关于非平稳高斯随机场的峰值高度分布：一维一般协方差与尺度空间

On the peak height distribution of non-stationary Gaussian random fields: 1D general covariance and scale space

本文推导一维一般协方差非平稳高斯过程的显式峰值高度分布，并推广至多维尺度空间场，提出两种高效数值算法。

Strongly mixing smooth planar vector field without asymptotic directions

We use a Voronoi-type tessellation based on a compound Poisson point process to construct a polynomially mixing stationary random smooth planar vector field with bounded nonnegative components such that, with probability one, none of the associated integral curves possess an asymptotic direction.

Integrability and Approximability of Solutions to the Stationary Diffusion Equation with Lévy Coefficient

We investigate the stationary diffusion equation with a coefficient given by a (transformed) Lévy random field. Lévy random fields are constructed by smoothing Lévy noise fields with kernels from the Matérn class. We show that Lévy noise naturally extends Gaussian white noise within Minlos' theory of generalized random fields. Results on the distributional path spaces of Lévy noise are derived as well as the amount of smoothing to ensure such distributions become continuous paths. Given this, we derive results on the pathwise existence and measurability of solutions to the random boundary valu…

Regularity of the stationary density for systems with fast random switching

We consider the piecewise-deterministic Markov process obtained by randomly switching between the flows generated by a finite set of smooth vector fields on a compact set. We obtain Hörmander-type conditions on the vector fields guaranteeing that the stationary density is: $C^k$ whenever the jump rates are sufficiently fast, for any $k<\infty$; unbounded whenever the jump rates are sufficiently slow and lower semi-continuous regardless of the jump rates. Our proofs are probabilistic, relying on a novel application of stopping times.