Strong Gaussian approximation for cumulative processes

Partial stochastic dominance for the multivariate Gaussian distribution

Gaussian comparison inequalities provide a way of bounding probabilities relating to multivariate Gaussian random vectors in terms of probabilities of random variables with simpler correlation structures. In this paper, we establish the partial stochastic dominance result that the cumulative distribution function of the maximum of a multivariate normal random vector, with positive intraclass correlation coefficient, intersects the cumulative distribution function of a standard normal random variable at most once. This result can be applied to the Bayesian design of a clinical trial in which se…

一个关于独立伽马随机变量之和及高斯变量二次型的分布函数在$(0,π)$上的积分

An integral over $(0,π)$ for the distribution function of a sum of independent gamma random variables and for quadratic forms of Gaussian variables

本文提出了独立伽马随机变量之和的累积分布函数在$(0,π)$上的积分表达式，并推导了正定二次型作为特例的结果。

高斯累积前景理论

Gaussian Cumulative Prospect Theory

本文提出一种新的累积前景理论参数化方法，使高斯奖励分布的赌博估值可用闭式公式计算，并兼顾理论有效性与表达丰富性，适用于大规模人群建模。

Non-asymptotic control of the cumulative distribution function of Lévy processes

We propose non-asymptotic controls of the cumulative distribution function $P(|X_{t}|\ge \varepsilon)$, for any $t>0$, $\varepsilon>0$ and any Lévy process $X$ such that its Lévy density is bounded from above by the density of an $α$-stable type Lévy process in a neighborhood of the origin. The results presented are non-asymptotic and optimal, they apply to a large class of Lévy processes.

局部自相似高斯过程的逗留时间

Sojourns of locally self-similar Gaussian processes

本文针对局部自相似高斯过程，推导了有限时间区间上累积巴黎破产概率的精确渐近近似，并证明Berman常数可由不同于分数布朗运动的自相似过程定义。

On the superposition of heterogeneous traffic at large time scales

Various empirical and theoretical studies indicate that cumulative network traffic is a Gaussian process. However, depending on whether the intensity at which sessions are initiated is large or small relative to the session duration tail, Mikosch et al. (Ann Appl Probab, 12:23-68, 2002) and Kaj and Taqqu (Progress Probab, 60:383-427, 2008) have shown that traffic at large time scales can be approximated by either fractional Brownian motion (fBm) or stable Levy motion. We study distributional properties of cumulative traffic that consists of a finite number of independent streams and give an ex…

Integro-differential equations linked to compound birth processes with infinitely divisible addends

Stochastic modelling of fatigue (and other material's deterioration), as well as of cumulative damage in risk theory, are often based on compound sums of independent random variables, where the number of addends is represented by an independent counting process. We consider here a cumulative model where, instead of a renewal process (as in the Poisson case), a linear birth (or Yule) process is used. This corresponds to the assumption that the frequency of \textquotedblleft damage" increments accelerates according to the increasing number of \textquotedblleft damages". We start from the partial…

泊松簇过程泛函的样本路径大偏差

Sample-Path Large Deviations for Functionals of Poisson Cluster Processes

本文在联合正则变化假设下，为带标记泊松簇过程的中心化累积泛函建立了 M1 拓扑 Skorokhod 空间中的样本路径大偏差原理，将隐藏正则变化结果推广到混合二项泊松簇过程和 Hawkes 过程等重尾簇过程。

Tail asymptotics for cumulative processes sampled at heavy-tailed random times with applications to queueing models in Markovian environments

This paper considers the tail asymptotics for a cumulative process $\{B(t); t \ge 0\}$ sampled at a heavy-tailed random time $T$. The main contribution of this paper is to establish several sufficient conditions for the asymptotic equality ${\sf P}(B(T) > bx) \sim {\sf P}(M(T) > bx) \sim {\sf P}(T>x)$ as $x \to \infty$, where $M(t) = \sup_{0 \le u \le t}B(u)$ and $b$ is a certain positive constant. The main results of this paper can be used to obtain the subexponential asymptotics for various queueing models in Markovian environments. As an example, using the main results, we derive subexponen…

Time to Extinction in Subcritical Two-Sex Branching Processes

Lower and upper bounds for the cumulative distribution function (cdf) of the time to extinction in a subcritical two-sex branching process are derived. A recursive procedure for approximating this cdf is also utilized. The results are illustrated with some simulations.

补充过程的比率极限定理

Ratio limit theorem for renewal processes

本文研究了补充过程中非失败概率的比率极限性质，证明其收敛于补充时刻为零的概率，并探讨了收敛速率与分布正则性之间的关系。

Quantifying and managing uncertainty in piecewise-deterministic Markov processes

In piecewise-deterministic Markov processes (PDMPs) the state of a finite-dimensional system evolves continuously, but the evolutive equation may change randomly as a result of discrete switches. A running cost is integrated along the corresponding piecewise-deterministic trajectory up to the termination to produce the cumulative cost of the process. We address three natural questions related to uncertainty in cumulative cost of PDMP models: (1) how to compute the Cumulative Distribution Function (CDF) of the cumulative cost when the switching rates are fully known; (2) how to accurately bound…

First exit times from a bounded interval for Lévy processes

In this paper we study the mean of the first exit time from a bounded interval of various Lévy processes. We establish sharp two-sided estimates of the mean for Lévy processes under certain condition on their characteristic exponents. We also study the cumulative distribution function of the supremum and infimum processes. Finally, we establish integral conditions that assure that the renewal function of the ladder height process is comparable with the linear one.

Expected signature of Gaussian processes with strictly regular kernels

We compute the expected signature of a class of Gaussian processes which is a subclass of the Gaussian processes with regular kernels, in the sense of Alos, Mazet and Nualart.

Exact confidence intervals of the extended Orey index for Gaussian processes

In this paper exact confidence intervals for the Orey index of Gaussian processes are obtained using concentration inequalities for Gaussian quadratic forms and discrete observations of the underlying process. The obtained result is applied to Gaussian processes with the Orey index which not necessarily have stationary increments.

Large deviations for conditionally Gaussian processes: estimates of level crossing probability

The problem of (pathwise) large deviations for conditionally continuous Gaussian processes is investigated. The theory of large deviations for Gaussian processes is extended to the wider class of random processes -- the conditionally Gaussian processes. The estimates of level crossing probability for such processes are given as an application.

高斯过程和Parseval框架的展开

Expansions for Gaussian processes and Parseval frames

本文导出了巴拿赫空间中高斯随机向量的级数展开式与其再生核希尔伯特空间中Parseval标架之间的精确联系。将所得结果应用于路径连续高斯过程，得到了分数阶Ornstein-Uhlenbeck过程的一个新的最优展开式。最后将这一结果推广到具有凸协方差函数的高斯平稳过程。

Representation of self-similar Gaussian processes

We develop the canonical Volterra representation for a self-similar Gaussian process by using the Lamperti transformation of the corresponding stationary Gaussian process, where this latter one admits a canonical integral representation under the assumption of pure non-determinism. We apply the representation obtained for the self-similar Gaussian process to derive an expression for Gaussian processes that are equivalent in law to the self-similar Gaussian process in question.

Asymptotic behavior for an additive functional of two independent self-similar Gaussian processes

We derive the asymptotic behavior for an additive functional of two independent self-similar Gaussian processes when their intersection local time exists, using the method of moments.

Integration-by-Parts Characterizations of Gaussian Processes

The Malliavin integration-by-parts formula is a key ingredient to develop stochastic analysis on the Wiener space. In this article we show that a suitable integration-by-parts formula also characterizes a wide class of Gaussian processes, the so-called Gaussian Fredholm processes.

Gaussian过程的弱近似：从再生成过程出发

Weak approximation for Gaussian processes from renewal processes

本文研究了从再生成过程构建高斯过程近似的方法，证明了某些进程可用于近似分数布朗运动等，并提供了确保收敛的充分条件。

若干去均值高斯过程的$L_2$-小球渐近性

$L_2$-small ball asymptotics for some demeaned Gaussian processes

该研究探讨Green高斯过程的中心化操作，推导了若干去均值高斯过程的$L_2$-小球渐近性质，丰富了相关过程的渐近理论结果。

On Dümbgen's exponentially modified Laplace continued fraction for Mill's ratio

The approximation of the Gaussian cumulative distribution or of the related Mills ratio have a long history starting with Gauss and Laplace and continuing nowadays. Below, we improve an important family of bounds provided recently by Dümbgen.

带复合泊松跳的高斯Volterra过程的预测

Prediction of Gaussian Volterra Processes with Compound Poisson Jumps

该研究针对带复合泊松跳的高斯Volterra过程，推导得到了其预测规律。

Comparison theorems for the small ball probabilities of Gaussian processes in weighted $L_2$-norms

We prove comparison theorems for small ball probabilities of the Green Gaussian processes in weighted $L_2$-norms. We find the sharp small ball asymptotics for many classical processes under quite general assumptions on the weight.

增强高斯过程的大偏差原理

Large Deviation Principle for Enhanced Gaussian Processes

研究提升到N步自由幂零群群的高斯过程的大偏差原理。我们将其应用于一大类高斯过程解除几何粗糙路径。在Hoelder-或模拓扑中，增强（分数阶）布朗运动的大偏差原理作为特例出现.

粗糙Heston模型作为双变量累积重尾INAR过程的标度极限：弱误差界和期权定价

Rough Heston model as the scaling limit of bivariate cumulative heavy-tailed INAR processes: Weak-error bounds and option pricing

研究双变量累积重尾INAR(∞)过程的近不稳定特性，证明其在特定参数化和标度下收敛于粗糙Heston模型，推导出INAR不对称参数与极限波动过程杠杆相关性和扩散尺度的联系，并通过离散方程与连续分数Riccati方程比较，获得弱误差界和期权定价方法。

高斯过程和位移高斯过程的σ场微分

Differentiating sigma-fields for Gaussian and shifted Gaussian processes

我们在（可能漂移的）高斯过程的一般框架下研究了可微分和不可微分σ-场的概念，并刻画了它们在概率测度等价变化下的不变性.作为应用，我们研究了一类与移位分数布朗运动相关的随机导数。最后，我们建立了一个共同可测版本的微分过程的存在条件，我们概述了随机嵌入方程的一般框架。

Partially Gaussian Stationary Stochastic Processes in Discrete Time

We present here an elementary example, for every fixed positive integer $k,$ of a strictly stationary nongaussian stochastic process in discrete time, all of whose $k$-marginals are gaussian.

On estimation of the Orey index for a class of Gaussian processes

Orey suggested the definition of some index for Gaussian processes with stationary increments which determines various properties of the sample paths of this process. We give an extension of the definition of the Orey index for a second order stochastic processes which may not have stationary increments and estimate the Orey index for Gaussian process from discrete observations of its sample paths.

On a covariance structure of some subset of self-similar Gaussian processes

We introduce a class of self-similar Gaussian processes and provide sufficient and necessary conditions for a member of the class to admit a unique small scale limit in the Skorokhod space. The class includes several well known processes. An example of application to the problem of estimation is given.

Some superconcentration inequalities for extrema of stationary Gaussian Processes

This note is concerned with concentration inequalities for extrema of stationary Gaussian processes. It provides non-asymptotic tail inequalities which fully reflect the fluctuation rate, and as such improve upon standard Gaussian concentration. The arguments rely on the hypercontractive approach developed by Chatterjee for superconcentration variance bounds. Some statistical illustrations complete the exposition.

受扰高斯过程的大偏差及若干退出概率的对数渐近估计

Large deviations for perturbed Gaussian processes and logarithmic asymptotic estimates for some exit probabilities

本文研究由满足大偏差原理的高斯过程经扰动得到的非高斯过程族的大偏差，并据此推导出半空间和象限退出概率的对数渐近估计。

A canonical form for Gaussian periodic processes

This article provides a representation theorem for a set of Gaussian processes; this theorem allows to build Gaussian processes with arbitrary regularity and to write them as limit of random trigonometric series. We show via Karhunen-Love theorem that this set is isometrically equivalent to l2. We then prove that regularity of trajectory path of anyone of such processes can be detected just by looking at decrease rate of l2 sequence associated to him via isometry.

Modulus of continuity for superlacunar trigonometric series and continuity of Gaussian stationary random processes

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.

Extremes of $α(t)$-Locally stationary Gaussian processes with non-constant variances

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.

Necessary and Sufficient Conditions for Hölder Continuity of Gaussian Processes

The continuity of Gaussian processes is extensively studied topic and it culminates in the Talagrand's notion of majorizing measures that gives complicated necessary and sufficient conditions. In this note we study the Hölder continuity of Gaussian processes. It turns out that necessary and sufficient conditions can be stated in a simple form that is a variant of the celebrated Kolmogorov-Čentsov condition.

Weak convergence towards two independent Gaussian processes from a unique Poisson process

We consider two independent Gaussian processes that admit a representation in terms of a stochastic integral of a deterministic kernel with respect to a standard Wiener process. In this paper we construct two families of processes, from a unique Poisson process, the finite dimensional distributions of which converge in law towards the finite dimensional distributions of the two independent Gaussian processes. As an application of this result we obtain families of processes that converge in law towards fractional Brownian motion and sub-fractional Brownian motion.

Pfaffian点过程的高斯极限

Gaussian limit for Pfaffian point processes

本文证明了一类Pfaffian点过程线性统计量的中心极限定理，并应用于Sine_4和Sine_1过程，得到阶梯函数缩放线性统计量的高斯极限。

Filtering of Gaussian processes in Hilbert spaces

Linear filtering problem for infinite-dimensional Gaussian processes is studied, the observation process being finite-dimensional. Integral equations for the filter and for covariance of the error are derived. General results are applied to linear SPDEs driven by Gauss-Volterra process observed at finitely many points of the domain.

Parisian Ruin of Self-similar Gaussian Risk Processes

In this paper we derive the exact asymptotics of the probability of Parisian ruin for self-similar Gaussian risk processes. Additionally, we obtain the normal approximation of the Parisian ruin time and derive an asymptotic relation between the Parisian and the classical ruin times.

Limit properties of exceedances point processes of scaled stationary Gaussian sequences

We derive the limiting distributions of exceedances point processes of randomly scaled weakly dependent stationary Gaussian sequences under some mild asymptotic conditions. In the literature analogous results are available only for contracted stationary Gaussian sequences. In this paper, we include additionally the case of randomly inflated stationary Gaussian sequences with a Weibullian type random scaling. It turns out that the maxima and minima of both contracted and inflated weakly dependent stationary Gaussian sequences are asymptotically independent.

关于自相似Volterra高斯过程的遍历变换的注记

A note on ergodic transformations of self-similar Volterra Gaussian processes

本文导出了一类Volterra型自相似高斯过程的遍历变换，即X_t = int^t_0 z_X（t，s）dW_s，t>0，其中z_X是确定核，W是标准布朗运动.

Gaussian limit for determinantal point processes with $J$-Hermitian kernels

We show that the central limit theorem for linear statistics over determinantal point processes with $J$-Hermitian kernels holds under fairly general conditions. In particular, We establish Gaussian limit for linear statistics over determinantal point processes on union of two copies of $\mathbb{R}^d$ when the correlation kernels are $J$-Hermitian translation-invariant.

On the maxima of continuous and discrete time Gaussian order statistics processes

In this paper, we study the asymptotic relation between the maximum of acontinuous order statistics process formed by stationary Gaussian processesand the maximum of this process sampled at discrete time points. It is shown that, these two maxima are asymptotically independent when the Gaussian processes are weakly dependent and the discrete points are sufficient sparse, while for other case, these two maxima are asymptotically dependent.

Functional Limit Theorems for Toeplitz Quadratic Functionals of Continuous time Gaussian Stationary Processes

\noindent The paper establishes weak convergence in $C[0,1]$ of normalized stochastic processes, generated by Toeplitz type quadratic functionals of a continuous time Gaussian stationary process, exhibiting long-range dependence. Both central and non-central functional limit theorems are obtained.

选定Volterra高斯过程的准螺旋性质

Quasihelix properties of selected Volterra Gaussian processes

本文研究了几种具有温控、幂加权和对数核的高斯Volterra过程的局部准螺旋和广义准螺旋性质，包括温控分数布朗运动和广义分数布朗运动型过程，探讨了参数值对性质的影响。

Local Times of Gaussian Processes

The aim of this work is to define and perform a study of local times of all Gaussian processes that have an integral representation over a real interval (that maybe infinite). Very rich, this class of Gaussian processes, contains Volterra processes (and thus fractional Brownian motion), multifractional Brownian motions as well as processes, the regularity of which varies along the time. Using the White Noise-based anticipative stochastic calculus with respect to Gaussian processes developed in [Leb17], we first establish a Tanaka formula. This allows us to define both weighted and non-weighted…

一般扰动风险过程的破产概率及其分解

Ruin probabilities and decompositions for general perturbed risk processes

研究了一类具有有限期望的服从子模型的带扰动的风险过程，其中扰动是一个零期望的谱负Levy过程.我们推导出一个Pollaczek-Hinchin型公式的生存概率的风险过程，并给出了一个解释的基础上分解的对偶风险过程在修改的阶梯时期的公式。

Ito's Formula for Gaussian Processes with Stochastic Discontinuities

We introduce a Skorokhod type integral and prove an Ito formula for a wide class of Gaussian processes which may exhibit stochastic discontinuities. Our Ito formula unifies and extends the classical one for general (i.e., possibly discontinuous) Gaussian martingales in the sense of Ito integration and the one for stochastically continuous Gaussian non-martingales in the Skorokhod sense, which was first derived in Alos et al. (Ann. Probab. 29, 2001).

平稳高斯过程类对数函数的中偏差

Moderate deviations for log-like functions of stationary Gaussian processes

建立了高斯过程非线性函数的中偏差原理。非线性函数不必是局部有界的。特别是，允许对数。(Thus，过程的小偏差是相关的。）离散时间和连续时间都被处理。假设相关函数具有可积幂型衰减。

对称简单排它过程的高斯估计

Gaussian estimates for symmetric simple exclusion processes

我们证明高斯尾估计的$n$粒子演变为对称排斥过程$\bb Z^d$，改进\cite{l}中得到的结果。从这一结果出发，我们导出了一维对称简单排它过程的一个非平衡Boltzmann-Gibbs原理，该原理从一个慢变参数的积测度出发.

Local times for multifractional square Gaussian processes

We consider multifractional process given by double Ito--Wiener integrals, which generalize the multifractional Rosenblatt process. We prove that this process is continuous and has a square integrable local time.

Lipschitz开集中具有高斯分量的各向同性Levy过程的Dirichlet热核估计

Dirichlet heat kernel estimates for isotropic Levy processes with Gaussian components in Lipschitz open sets

针对含高斯分量的不连续各向同性Levy过程，研究其在Lipschitz开集中的Dirichlet热核估计，给出双侧界及Varopoulos型估计成立的充要条件。