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This book presents recent methods of study on the asymptotic behavior of solutions of abstract differential equations such as stability, exponential dichotomy, periodicity, almost periodicity, and almost automorphy of solutions. The chosen methods are described in a way that is suitable to those who have some experience with ordinary differential equations. The book is intended for graduate students and researchers in the related areas.

Differential equations --- Asymptotic distribution (Probability theory) --- Asymptotic expansions --- Central limit theorem --- Distribution (Probability theory) --- 517.91 Differential equations --- Asymptotic theory.

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This textbook is devoted to the general asymptotic theory of statistical experiments. Local asymptotics for statistical models in the sense of local asymptotic (mixed) normality or local asymptotic quadraticity make up the core of the book. Numerous examples deal with classical independent and identically distributed models and with stochastic processes. The book can be read in different ways, according to possibly different mathematical preferences of the reader. One reader may focus on the statistical theory, and thus on the chapters about Gaussian shift models, mixed normal and quadratic models, and on local asymptotics where the limit model is a Gaussian shift or a mixed normal or a quadratic experiment (LAN, LAMN, LAQ). Another reader may prefer an introduction to stochastic process models where given statistical results apply, and thus concentrate on subsections or chapters on likelihood ratio processes and some diffusion type models where LAN, LAMN or LAQ occurs. Finally, readers might put together both aspects. The book is suitable for graduate students starting to work in statistics of stochastic processes, as well as for researchers interested in a precise introduction to this area.

Mathematical statistics --- Asymptotic distribution (Probability theory) --- Asymptotic expansions --- Central limit theorem --- Distribution (Probability theory) --- Asymptotic theory. --- General Asymptotic Theory. --- LAMN. --- LAN. --- LAQ. --- Local Asymptotic Normality. --- Local Asymptotic Quadraticity. --- Local Asymptotic. --- Statistical Experiment. --- Statistical Model.

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This study aims to embed the history of the central limit theorem within the history of the development of probability theory from its classical to its modern shape, and, more generally, within the corresponding development of mathematics. The history of the central limit theorem is not only expressed in light of "technical" achievement, but is also tied to the intellectual scope of its advancement. The history starts with Laplace's 1810 approximation to distributions of linear combinations of large numbers of independent random variables and its modifications by Poisson, Dirichlet, and Cauchy, and it proceeds up to the discussion of limit theorems in metric spaces by Donsker and Mourier around 1950. This self-contained exposition additionally describes the historical development of analytical probability theory and its tools, such as characteristic functions or moments. The importance of historical connections between the history of analysis and the history of probability theory is demonstrated in great detail. With a thorough discussion of mathematical concepts and ideas of proofs, the reader will be able to understand the mathematical details in light of contemporary development. Special terminology and notations of probability and statistics are used in a modest way and explained in historical context.

Central limit theorem -- History. --- Central limit theorem --- Mathematics --- Physical Sciences & Mathematics --- Mathematical Statistics --- History --- Central limit theorem. --- Free probability theory. --- Probability theory, Free --- Mathematics. --- History. --- Probabilities. --- Statistics. --- History of Mathematical Sciences. --- Probability Theory and Stochastic Processes. --- Statistics, general. --- Asymptotic distribution (Probability theory) --- Limit theorems (Probability theory) --- Operator algebras --- Selfadjoint operators --- Distribution (Probability theory. --- Statistical analysis --- Statistical data --- Statistical methods --- Statistical science --- Econometrics --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Statistics . --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Annals --- Auxiliary sciences of history --- Math --- Science --- Mathematics - History --- Distribution (Probability theory) --- Statistics

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The work of E. Hopf and G.A. Hedlund, in the 1930s, on transitivity and ergodicity of the geodesic flow for hyperbolic surfaces, marked the beginning of the investigation of the statistical properties and stochastic behavior of the flow. The first central limit theorem for the geodesic flow was proved in the 1960s by Y. Sinai for compact hyperbolic manifolds. Since then, strong relationships have been found between the fields of ergodic theory, analysis, and geometry. Different approaches and new tools have been developed to study the geodesic flow, including measure theory, thermodynamic formalism, transfer operators, Laplace operators, and Brownian motion. All these different points of view have led to a deep understanding of more general dynamical systems, in particular the so-called Anosov systems, with applications to geometric problems such as counting, equirepartition, mixing, and recurrence properties of the orbits. This book comprises two independent texts that provide a self-contained introduction to two different approaches to the investigation of hyperbolic dynamics. The first text, by S. Le Borgne, explains the method of martingales for the central limit theorem. This approach can be used in several situations, even for weakly hyperbolic flows, and the author presents a good number of examples and applications to equirepartition and mixing. The second text, by F. Faure and M. Tsujii, concerns the semiclassical approach, by operator theory: chaotic dynamics is described through the spectrum of the associated transfer operator, with applications to the asymptotic counting of periodic orbits. The book will be of interest for a broad audience, from PhD and Post-Doc students to experts working on geometry and dynamics.

Geodesic flows. --- Central limit theorem. --- Ergodic theory. --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Asymptotic distribution (Probability theory) --- Limit theorems (Probability theory) --- Flows (Differentiable dynamical systems) --- Differentiable dynamical systems. --- Distribution (Probability theory. --- Operator theory. --- Global differential geometry. --- Dynamical Systems and Ergodic Theory. --- Probability Theory and Stochastic Processes. --- Operator Theory. --- Hyperbolic Geometry. --- Differential Geometry. --- Geometry, Differential --- Functional analysis --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Dynamics. --- Probabilities. --- Hyperbolic geometry. --- Differential geometry. --- Differential geometry --- Hyperbolic geometry --- Lobachevski geometry --- Lobatschevski geometry --- Geometry, Non-Euclidean --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics