Prediction Theory for Stationary Functional Time Series

A Survey on Limit Theorems for Toeplitz Type Quadratic Functionals of Stationary Processes and Applications

This is a survey of recent results on central and non-central limit theorems for quadratic functionals of stationary processes. The underlying processes are Gaussian, linear or Lévy-driven linear processes with memory, and are defined either in discrete or continuous time. We focus on limit theorems for Toeplitz and tapered Toeplitz type quadratic functionals of stationary processes with applications in parametric and nonparametric statistical estimation theory. We discuss questions concerning Toeplitz matrices and operators, Fejér-type singular integrals, and Lévy-Itô-type and Stratonovich-ty…

A Many-Server Functional Strong Law For A Non-Stationary Loss Model

The purpose of this note is to show that it is possible to establish a many-server functional strong law of large numbers (FSLLN) for the fraction of occupied servers (i.e., the scaled number-in-system) without explicitly tracking either the age or the residual service times of the jobs in a non-Markovian, non-stationary loss model. This considerable analytical simplification is achieved by exploiting a semimartingale representation. The fluid limit is shown to be the unique solution of a Volterra integral equation.

平稳过程泛函的一个四阶矩不等式

A fourth moment inequality for functionals of stationary processes

本文建立了强遍历马尔可夫链泛函部分和的四阶矩界。这类不等式在经验过程不变原理的研究中起着重要的作用。这一个是特别适应的技术Dehling，Durieu和Volnnesi（2008）。对于传递算子具有某些谱性质的动力系统，可以证明同样的矩界。给出了应用实例。

保守流生成的负相依重尾平稳无穷可分过程的泛函中心极限定理

Functional central limit theorem for negatively dependent heavy-tailed stationary infinitely divisible processes generated by conservative flows

证明了一类负相依的对称平稳长程相依重尾无穷可分过程部分和的泛函中心极限定理。以前只有积极的依赖可以治疗。负相关性涉及高斯二阶的抵消。这导致了新类型的限制过程，涉及稳定的随机措施，由于重尾，米塔格-莱弗勒过程，由于长记忆，和布朗运动，由于高斯二阶取消。

关于起始于一点的平稳马尔可夫链的泛函CLT

On the functional CLT for stationary Markov Chains started at a point

我们提出了一个一般的功能中心极限定理开始于一个点，也称为下淬火。因此，我们指出了几个新的类的平稳过程，通过投影条件，满足这种类型的渐近结果定义。其中一个定理表明，如果马尔可夫链是平稳遍历可逆的，则该结果对该链的有界可加泛函在L_1表示中具有鞅上边界时成立。我们的结果也很好地适应强混合序列提供这种情况下的替代，更短的方法，最近在文献中的一些结果。

具有无限延迟的随机泛函微分方程 stationary 解的 large deviation 原理

Large deviation principle for the stationary solutions of stochastic functional differential equations with infinite delay

本文研究了具有无限延迟的随机泛函微分方程 stationary 解的 LDP，在小扰动下证明了解的存在性、唯一性及解映射的均匀 LDP，并通过收缩原理获得不变测度的 LDP。

Functional CLT for martingale-like nonstationary dependent structures

In this paper we develop non-stationary martingale techniques for dependent data. We shall stress the non-stationary version of the projective Maxwell-Woodroofe condition, which will be essential for obtaining maximal inequalities and functional central limit theorem for the following examples: nonstationary \r{ho}-mixing sequences, functions of linear processes with non-stationary innovations, quenched version of the functional central limit theorem for a stationary sequence, evolutions in random media such as a process sampled by a shifted Markov chain.

具有平稳复高斯系数的随机幂级数的零点密度

The density of zeros of random power series with stationary complex Gaussian coefficients

研究具有平稳复高斯系数的随机幂级数，分析了收敛圆边界附近零点径向密度的渐近行为，揭示了系数依赖性降低零点密度的机制及谱测度支撑与解析延拓的关系。

保守流生成的重尾平稳无穷可分过程的泛函中心极限定理

Functional central limit theorem for heavy tailed stationary infinitely divisible processes generated by conservative flows

对一类具有正则变Lévy测度的对称平稳无穷可分过程的部分和建立了一类新的泛函中心极限定理.极限过程是一类新的具有平稳增量的对称稳定自相似过程，它的一部分参数空间与先前描述的过程重合。规范化序列和限制过程是由过程的整表示下的流的遍历理论性质决定的。这些性质可以解释为决定了平稳无限可分过程的记忆长度。我们还建立了函数收敛，在一个强分布意义下，保守的逐点对偶遍历映射保持一个无限的措施。

平稳点过程的非局部Wasserstein几何、梯度流与泛函不等式

Non-local Wasserstein Geometry, Gradient Flows, and Functional Inequalities for Stationary Point Processes

该研究在平稳点过程空间构造非局部Benamou-Brenier型传输距离，证明其为[DSHS24]中传输距离的变体，揭示Ornstein-Uhlenbeck半群是相对熵的梯度流，还建立了平稳测地线存在性、相对熵的1-测地凸性及泛函不等式的平稳类似物。

在混合时间之前和之后：关于平稳高斯-马尔可夫过程加性函数的新的近似方法

Before and beyond the mixing time: New approximations for additive functionals of stationary Gauss-Markov processes

本文研究了平稳高斯-马尔可夫过程加性函数在观测数量与混合时间比例不同时的近似问题，提出了一种新的定量和函数极限定理，揭示了在混合时间附近出现的相变现象及新极限过程的性质。

A 2-Dimensional Functional Central Limit Theorem for Non-stationary Dependent Random Fields

We obtain an elementary invariance principle for multi-dimensional Brownian sheet where the underlying random fields are not necessarily independent or stationary. Possible applications include unit-root tests for spatial as well as panel data models.

具有平稳系数的高斯幂级数零点相关函数的谱表示

Spectral representation of correlation functions for zeros of Gaussian power series with stationary coefficients

本文研究系数为离散平稳高斯过程的高斯解析函数，利用谱测度重新考察极限定理、数值检验例子，并给出零点集n点相关函数的积分表示。

关于平稳马尔可夫控制的长期泛函稳定性

Stability of long run functionals with respect to stationary Markov controls

本文研究平稳马尔可夫控制逐点收敛时长期泛函的稳定性，考虑单位时间平均成本与长期风险敏感两类泛函，通过一致遍历性假设及其放宽，结合受控转移概率收敛性分析其依赖性。

On the the linear processes of a stationary time series AR(2)

Our aim in this work is to give explicit formula of the linear processes solution of autoregressive time series AR(2) with hint of generating functions theory by using the Horadam numbers and polynomials.

SPHARMA approximations for stationary functional time series on the sphere

In this paper, we focus on isotropic and stationary sphere-cross-time random fields. We first introduce the class of spherical functional autoregressive-moving average processes (SPHARMA), which extend in a natural way the spherical functional autoregressions (SPHAR) recently studied in [8, 7]; more importantly, we then show that SPHAR and SPHARMA processes of sufficiently large order can be exploited to approximate every isotropic and stationary sphere-cross-time random field, thus generalizing to this infinite-dimensional framework some classical results on real-valued stationary processes. …

平稳马氏过程占据时间泛函的估计误差

Estimation error for occupation time functionals of stationary Markov processes

研究了平稳马尔可夫过程的积分泛函的Riemann和估计的逼近问题。平稳性和过程的无穷小生成元的谓词演算被用来更好地理解估计误差，并证明了一般的误差界。所提出的方法承认一般的被积函数，并给出了一个统一的解释不同的利率在文献中得到。几个例子演示了如何一般的界限可以与众所周知的函数空间。

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。

遍历平稳时间序列的非参数推断

Nonparametric inference for ergodic, stationary time series

该设置是平稳的、遍历的时间序列。挑战在于构造一个函数序列，每个函数都只基于过去的有限段，在给定无限过去的情况下，这些函数一起为下一个观测的条件概率提供强一致的估计。Ornstein给出了这样一个建设的情况下，价值观是从一个有限集，最近Algoet扩展计划的时间序列坐标在波兰空间。本研究报告提出了一种不同的解决办法来应对这一挑战。该算法简单，验证相当透明。回归，模式识别和在线预测的一些扩展被提到。

Central Limit Theorems for a Stationary Semicircular Sequence in Free Probability

In this paper, we focus on studying central limit theorems for functionals of some specific stationary random processes. In classical probability theory, it is well-known that for non-linear functionals of stationary Gaussian sequences, we can get a central-limit result via Hermite polynomials and the diagram formula for cumulants. In this paper, the main result is an analogous central limit theorem, in a free probability setting, for real-valued functionals of a stationary semicircular sequence with long-range dependence, namely the correlation function of the underlying time series tends to …

先验联合分布平稳时间序列综合的矩阵乘积法

Matrix products for the synthesis of stationary time series with a priori prescribed joint distributions

从非平衡统计物理模型的启发，一个通用的框架，使固定的时间序列的定义和合成的先验规定和控制的联合分布。它的中心特征包括为联合分布保留其在独立性下具有的简单产品结构，同时能够在样本之间输入受控和规定的依赖性。为此，它基于d维矩阵的乘积，其条目由有效分布组成。由此定义的时间序列的统计特性进行了详细研究。已经能够重铸这个框架到隐马尔可夫模型，使我们能够获得一个有效的合成过程。教学精心挑选的例子（时间序列具有相同的边缘分布，相同的协方差函数，但不同的联合分布）旨在说明该方法的力量和潜力，并显示如何有针对性的统计特性，可以实际规定。

平稳时间序列的间歇估计

Intermittent estimation of stationary time series

设$\{X_n\}_{n=0}^{\infty}$是具有未知分布的平稳实值时间序列。我们的目标是以强一致的方式基于观测$X_i$，$0\le i\le n$估计$X_{n+1}$的条件期望。Bailey和Ryabko证明了，即使遍历二值时间序列，如果在所有值的$n$估计，这是不可能的。我们提出了一个非常简单的算法，它将使预测无限经常在精心选择的停时选择我们的规则。我们证明了在一定条件下，我们的程序是强（逐点）一致的，和$L_2$一致的没有任何条件。本文还给出了该系统停时增长的上界。

Regular multidimensional stationary time series

The aim of this paper is to give a simpler, more usable sufficient condition to the regularity of generic weakly stationary time series. Also, this condition is used to show how regular processes satisfying these sufficient conditions can be approximated by a lower rank \emph{regular} process. The relevance of these issues is shown by the ever increasing presence of high-dimensional data in many fields lately, and because of this, low rank processes and low rank approximations are becoming more important. Moreover, regular processes are the ones which are completely influenced by random innova…

平稳二值时间序列的预测

Forecasting for stationary binary time series

平稳和遍历的二值时间序列$\{X_n\}_{n=0}^{\infty}$的预测问题是在没有过程$\{X_n\}$的分布的先验知识的情况下，基于观测$X_i$、$0\le i\le n$来估计$X_{n+1}=1$的概率。众所周知，如果在$n$的所有值处进行估计，则这是不可能的。我们提出了一个简单的程序，将试图使这样的预测无限经常在精心选择的停时选择的算法。我们证明了所提出的方法在一定条件下是一致的，并且我们估计了停时的增长率。

Continuous-time locally stationary time series models

We adapt the classical definition of locally stationary processes in discrete-time to the continuous-time setting and obtain equivalent representations in the time and frequency domain. From this, a unique time-varying spectral density is derived using the Wigner-Ville spectrum. As an example, we investigate time-varying Lévy-driven state space processes, including the class of time-varying Lévy-driven CARMA processes. First, the connection between these two classes of processes is examined. Considering a sequence of time-varying Lévy-driven state space processes, we then give sufficient condi…

离散时间序列的预测

Prediction for discrete time series

设$\{X_n\}$是从有限或可数无限集合${\cal X}$中取值的平稳遍历时间序列。假设过程的分布是未知的。我们提出了一个序列的停时$λ_n$沿，我们将能够估计的条件概率$P(X_{λ_n+1}=x|X_0,...,X_{λ_n})$从数据段$(X_0,...,X_{λ_n})$在逐点一致的方式为一个限制类的固定和遍历有限或可数无限字母时间序列，其中包括除其他外，所有的固定和遍历finitarily马尔可夫过程。如果平稳和遍历过程最终证明是马尔可夫的（除其他外，所有平稳和遍历马尔可夫链都包括在这个类中），那么$ \lim_{n\to \infty} {n\over λ_n}>0$几乎肯定。如果平稳遍历过程具有有限熵率，那么$λ_n$的上界是多项式的，最终几乎肯定。

A dependent Lindeberg central limit theorem for cluster functionals on stationary random fields

In this paper, we provide a central limit theorem for the finite-dimensional marginal distributions of empirical processes $(Z_n(f))_{f\in\mathcal{F}}$ whose index set $\mathcal{F}$ is a family of cluster functionals valued on blocks of values of a stationary random field. The practicality and applicability of the result depends mainly on the usual Lindeberg condition and a sequence $T_n$ which summarizes the dependence between the blocks of the random field values. Finally, as application, we use the previous result in order to show the Gaussian asymptotic behavior of the iso-extremogram esti…

Non-parametric estimation of Stochastic Differential Equations from stationary time-series

We study efficiency of non-parametric estimation of diffusions (stochastic differential equations driven by Brownian motion) from long stationary trajectories. First, we introduce estimators based on conditional expectation which is motivated by the definition of drift and diffusion coefficients. These estimators involve time- and space-discretization parameters for computing expected values from discretely-sampled stationary data. Next, we analyze consistency and mean squared error of these estimators depending on computational parameters. We derive relationships between the number of observa…

平稳二值时间序列的输出猜测

Guessing the output of a stationary binary time series

二值时间序列$\{X_n\}_{n=0}^{\infty}$的前向预测问题是在没有过程$\{X_n\}$的分布的先验知识的情况下，基于观测$X_i$、$0\le i\le n$来估计$X_{n+1}=1$的概率。众所周知，如果在$n$的所有值处进行估计，则这是不可能的。我们提出了一个简单的程序，将试图使这样的预测无限经常在精心选择的停时选择的算法。还显示了停时的增长率。