Statistical Inference for Ergodic Diffusion Processes
Statistical Inference for Ergodic Diffusion Processes
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DOI：
10.1198/jasa.2006.s98
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发表时间：
2006-06
影响因子：
3.7
通讯作者：
P. Kiessler
中科院分区：
文献类型：
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作者：
P. Kiessler
“列维过程”（八篇论文），“III. 经验过程”（四篇论文），以及“IV. 随机微分方程”（四篇论文）。以下是对各篇论文的一些评论：在I.2（第一部分的第二篇论文）中，基于路径空间上的克拉克公式的协方差表示法被用于获得流形上布朗运动泛函的集中不等式，从而能够得到该布朗运动的尾估计。在I.4中，得到了\(R\)中规范高斯测度的一个运输不等式，并将其应用于具有非对称伯努利系数的随机级数范数的欣钦 - 卡汉不等式。在II.1中，给出了二阶\(U\)-统计量的指数不等式；这些不等式依赖于经验过程的塔拉格兰不等式，但也使用了鞅型不等式。在II.2中，研究了巴拿赫空间中一个高斯[以及更一般的，独立同分布（iid）]级数的无条件收敛性。给出了对高斯过程的卡亨南 - 洛芙表示的应用。在II.3中，给出了由具有对数凹尾的正随机变量生成的多维混沌的尾性质和矩的估计。在II.4中，提出了一种用于研究无限维马尔可夫过程序列渐近分布的定量技术。证明依赖于相关的指数鞅序列的性质。在II.5中，表明由一个对称列维过程驱动且核具有有限总2 - 变差的移动平均过程存在一个几乎必然有界的版本。在II.2中，提出了一种在中心极限定理中研究熵收敛的马尔可夫方法。重点在于收敛速度，以及放宽一个谱隙假设。在II.7中，借助超压缩方法给出了一般伯努利随机变量的欣钦 - 卡汉不等式的一个新版本。在III.1中，给出了可分巴拿赫空间上经验过程和独立同分布随机向量和的中偏差的充分必要条件。在III.2中，得到了独立随机变量的次可加函数的指数集中不等式。因此，由于M. 勒杜克斯引入的熵方法的进一步发展，经验过程的塔拉格兰不等式得到了改进。在III.3中，借助集中不等式得到了经验过程的比率极限定理。在III.4中，通过均匀经验过程的加权逼近结果得到了真实分布函数和经验分布函数之间截尾瓦瑟斯坦距离的渐近分布。在IV.1中，得到了随机偏微分方程分裂逼近的精确收敛速度。误差是根据索伯列夫范数估计的。在IV.4中，得到了由赫斯特指数\(H < 1/2\)的分数布朗运动驱动且具有可能依赖于时间的漂移（满足一个合适的可积性条件）的随机微分方程强解的存在性和唯一性。简而言之，这本书呈现了各种各样为随机过程或随机变量序列建立的不等式。在2006年，这仍然是运输问题研究的一个活跃领域。
Lévy Processes” (eight papers), “III. Empirical Processes” (four papers), and “IV. Stochastic Differential Equations” (four papers). Here are some comments about the individual papers: In I.2 (paper 2 of Part I) the covariance representation method, which relies on Clark’s formula on path spaces, is used to obtain concentration inequalities for functionals of Brownian motion on a manifold, allowing one to obtain tail estimates for this Brownian motion. In I.4 a transportation inequality for the canonical Gaussian measure in R is obtained and applied to Khintchine–Kahane inequalities for norms of random series with nonsymmetric Bernoulli coefficients. In II.1 exponential inequalities for U -statistics of order two are presented; these rely upon the Talagrand inequality for empirical processes but also use martingale type inequalities. In II.2 the unconditional convergence of a Gaussian [and, more generally, independent, identically distributed (iid)] series in a Banach space is studied. Applications to Karhunen–Love representations of Gaussian processes are given. In II.3 estimates of tail properties and moments of multidimensional chaos generated by positive random variables with log concave tails are given. In II.4 a quantitative technique for studying the asymptotic distribution of sequences of Markov processes in infinite dimensions is proposed. The proof relies on the properties of an associated sequence of exponential martingales. In II.5 it is shown that a moving average process driven by a symmetric Lévy process and with a kernel with finite total 2-variation admits an almost surely bounded version. In II.6 a Markovian approach to the entropic convergence in the central limit theorem is presented. The emphasis is on the speed of convergence, as well as relaxing a spectral gap assumption. In II.7 a new version of the Khintchine–Kahane inequality for general Bernoulli random variables is presented with the help of hypercontractive methods. In III.1 necessary and sufficient conditions for the moderate deviations of empirical processes and sums of iid random vectors on a separable Banach space are given. In III.2 exponential concentration inequalities for subadditive functions of independent random variables are obtained. As a consequence, Talagrand’s inequality for empirical processes is refined thanks to further developments of the entropy method introduced by M. Ledoux. In III.3 ratio limit theorems for empirical processes are obtained with the help of concentration inequalities. In III.4 asymptotic distributions of trimmed Wasserstein distances between the true and the empirical distribution function are obtained via weighted approximation results for uniform empirical processes. In IV.1 sharp rates of convergence for splitting-up approximations of stochastic partial differential equations are obtained. The error is estimated in terms of Sobolev’s norm. In IV.4 the existence and uniqueness of a strong solution for a stochastic differential equation driven by a fractional Brownian motion with Hurst index H < 1/2 and with a possibly time-dependent drift which satisfies a suitable integrability condition is obtained. In short, the book presents a wide variety of inequalities which are established for either stochastic processes or sequences of random variables. In 2006 this is still an active domain of research in transportation problems.