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Local Properties Of Distributions Of Stochastic Functionals

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### Local Properties of Distributions of Stochastic Functionals by Yu. A. Davydov, M. A. Lifshits, andN. V. Smorodina Book Summary:

This book investigates the distributions of functionals defined on the sample paths of stochastic processes. It contains systematic exposition and applications of three general research methods developed by the authors. (i) The method of stratifications is used to study the problem of absolute continuity of distribution for different classes of functionals under very mild smoothness assumptions. It can be used also for evaluation of the distribution density of the functional. (ii) The method of differential operators is based on the abstract formalism of differential calculus and proves to be a powerful tool for the investigation of the smoothness properties of the distributions. (iii) The superstructure method, which is a later modification of the method of stratifications, is used to derive strong limit theorems (in the variation metric) for the distributions of stochastic functionals under weak convergence of the processes. Various application examples concern the functionals of Gaussian, Poisson and diffusion processes as well as partial sum processes from the Donsker-Prokhorov scheme. The research methods and basic results in this book are presented here in monograph form for the first time. The text would be suitable for a graduate course in the theory of stochastic processes and related topics.

### Translations of Mathematical Monographs by N.A Book Summary:

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### Gaussian Random Functions by M.A. Lifshits Book Summary:

It is well known that the normal distribution is the most pleasant, one can even say, an exemplary object in the probability theory. It combines almost all conceivable nice properties that a distribution may ever have: symmetry, stability, indecomposability, a regular tail behavior, etc. Gaussian measures (the distributions of Gaussian random functions), as infinite-dimensional analogues of tht

### Mathematics of Information and Coding by Te Sun Han,Kingo Kobayashi Book Summary:

This book is intended to provide engineering and/or statistics students, communications engineers, and mathematicians with the firm theoretic basis of source coding (or data compression) in information theory. Although information theory consists of two main areas, source coding and channel coding, the authors choose here to focus only on source coding. The reason is that, in a sense, it is more basic than channel coding, and also because of recent achievements in source coding and compression. An important feature of the book is that whenever possible, the authors describe universal coding methods, i.e., the methods that can be used without prior knowledge of the statistical properties of the data. The authors approach the subject of source coding from the very basics to the top frontiers in an intuitively transparent, but mathematically sound, manner.

### Gaussian Measures by Vladimir I. Bogachev Book Summary:

This book gives a systematic exposition of the modern theory of Gaussian measures. It presents with complete and detailed proofs fundamental facts about finite and infinite dimensional Gaussian distributions. Covered topics include linear properties, convexity, linear and nonlinear transformations, and applications to Gaussian and diffusion processes. Suitable for use as a graduate text and/or a reference work, this volume contains many examples, exercises, and an extensive bibliography. It brings together many results that have not appeared previously in book form.

### Function Spaces in Analysis by Krzysztof Jarosz Book Summary:

This volume contains the proceedings of the Seventh Conference on Function Spaces, which was held from May 20-24, 2014 at Southern Illinois University at Edwardsville. The papers cover a broad range of topics, including spaces and algebras of analytic functions of one and of many variables (and operators on such spaces), spaces of integrable functions, spaces of Banach-valued functions, isometries of function spaces, geometry of Banach spaces, and other related subjects.

### Algebraic Geometry 1 by 健爾·上野 Book Summary:

Beginning algebraic geometers are well served by Uneno's inviting introduction to the language of schemes. Grothendieck's schemes and Zariski's emphasis on algebra and rigor are primary sources for this introduction to a rich mathematical subject. Ueno's book is a self-contained text suitable for an introductory course on algebraic geometry.

### Limit Theorems for Functionals of Random Walks by A. N. Borodin,Il'dar Abdulovich Ibragimov,Ilʹdar Abdulovich Ibragimov Book Summary:

This book examines traditional problems in the theory of random walks: limit theorems for additive and multiadditive functionals defined on a random walk. Although the problems are traditional, the methods presented here are new. One of the methods involves expressing the functionals in terms of a suitable integral transform (such as the Fourier transform), and another method is based on results on convergence of processes generated by random walks to Brownian local time. These methods can be used to prove the convergence of functionals of random walks under very general assumptions about the functional and for a very broad class of random walks. The book is intended for experts in probability theory and its applications, as well as for undergraduate and graduate students specializing in these areas.

### Journal of the American Statistical Association by N.A Book Summary:

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### Mathematical Reviews by N.A Book Summary:

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### Asymptotic Methods in the Theory of Gaussian Processes and Fields by Vladimir I. Piterbarg Book Summary:

This book is devoted to a systematic analysis of asymptotic behavior of distributions of various typical functionals of Gaussian random variables and fields. The text begins with an extended introduction, which explains fundamental ideas and sketches the basic methods fully presented later in the book. Good approximate formulas and sharp estimates of the remainders are obtained for a large class of Gaussian and similar processes. The author devotes special attention to the development of asymptotic analysis methods, emphasizing the method of comparison, the double-sum method and the method of moments. The author has added an extended introduction and has significantly revised the text for this translation, particularly the material on the double-sum method.

### Theory of Random Sets by Ilya Molchanov Book Summary:

This is the first systematic exposition of random sets theory since Matheron (1975), with full proofs, exhaustive bibliographies and literature notes Interdisciplinary connections and applications of random sets are emphasized throughout the book An extensive bibliography in the book is available on the Web at http://liinwww.ira.uka.de/bibliography/math/random.closed.sets.html, and is accompanied by a search engine

### Probability and Mathematical Statistics by N.A Book Summary:

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### Functional Equations and Characterization Problems on Locally Compact Abelian Groups by Gennadiĭ Mikhaĭlovich Felʹdman,Gennadij M. Fel'dman Book Summary:

This book deals with the characterization of probability distributions. It is well known that both the sum and the difference of two Gaussian independent random variables with equal variance are independent as well. The converse statement was proved independently by M. Kac and S. N. Bernstein. This result is a famous example of a characterization theorem. In general, characterization problems in mathematical statistics are statements in which the description of possible distributions of random variables follows from properties of some functions in these variables. In recent years, a great deal of attention has been focused upon generalizing the classical characterization theorems to random variables with values in various algebraic structures such as locally compact Abelian groups, Lie groups, quantum groups, or symmetric spaces. The present book is aimed at the generalization of some well-known characterization theorems to the case of independent random variables taking values in a locally compact Abelian group $X$. The main attention is paid to the characterization of the Gaussian and the idempotent distribution (group analogs of the Kac-Bernstein, Skitovich-Darmois, and Heyde theorems). The solution of the corresponding problems is reduced to the solution of some functional equations in the class of continuous positive definite functions defined on the character group of $X$. Group analogs of the Cramer and Marcinkiewicz theorems are also studied. The author is an expert in algebraic probability theory. His comprehensive and self-contained monograph is addressed to mathematicians working in probability theory on algebraic structures, abstract harmonic analysis, and functional equations. The book concludes with comments and unsolved problems that provide further stimulation for future research in the theory.

### Handbook of Stochastic Analysis and Applications by D. Kannan,V. Lakshmikantham Book Summary:

An introduction to general theories of stochastic processes and modern martingale theory. The volume focuses on consistency, stability and contractivity under geometric invariance in numerical analysis, and discusses problems related to implementation, simulation, variable step size algorithms, and random number generation.

### Probability Theory and Mathematical Statistics by B. Grigelionis Book Summary:

These Proceedings, published in two volumes contain the invited lectures and some selected short communications by international scientists which were presented at the Vth International Vilnius Conference on Probability Theory and Mathematical Statistics, Vilnius, Lithuania, 25 June--1 July 1989. Only available as a 2-volume set The main topics of presented papers cover the contempory techniques and results in limit theorems of probability and mathematical statistics, theory of Markov processes, martingales, random fields, stochastic physics, stochastic evolution equations, probability distributions on functional spaces and probabilistic number theory. In addition papers of leading specialists in quantum probability are included.

### Geometric Problems in the Theory of Infinite-dimensional Probability Distributions by V. N. Sudakov Book Summary:

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### Eigenvalue Distribution of Large Random Matrices by Leonid Andreevich Pastur,Mariya Shcherbina Book Summary:

Random matrix theory is a wide and growing field with a variety of concepts, results, and techniques and a vast range of applications in mathematics and the related sciences. The book, written by well-known experts, offers beginners a fairly balanced collection of basic facts and methods (Part 1 on classical ensembles) and presents experts with an exposition of recent advances in the subject (Parts 2 and 3 on invariant ensembles and ensembles with independent entries). The text includes many of the authors' results and methods on several main aspects of the theory, thus allowing them to present a unique and personal perspective on the subject and to cover many topics using a unified approach essentially based on the Stieltjes transform and orthogonal polynomials. The exposition is supplemented by numerous comments, remarks, and problems. This results in a book that presents a detailed and self-contained treatment of the basic random matrix ensembles and asymptotic regimes. This book will be an important reference for researchers in a variety of areas of mathematics and mathematical physics. Various chapters of the book can be used for graduate courses; the main prerequisite is a basic knowledge of calculus, linear algebra, and probability theory.

### The Pharmacology of Functional, Biochemical, and Recombinant Receptor Systems by Terrence P. Kenakin,James A. Angus Book Summary:

This, the 148th volume of the Handbook of Experimental Pharmacology series, focuses on the very core of pharmacology, namely receptor theory. It is fitting that the originator of receptor pharmacology, A.J. CLARK, authored the fourth volume of this series 63 years ago. In that volume CLARK further developed his version of receptor theory first described four years earlier in his classic book The Mode of Action of Drugs. An examination of the topics covered in volume 4 reveals a striking similarity to the topics covered in this present volume; pharmacologists today are still as interested in unlocking the secrets of dose-response relationships to reveal the biological and cheƯ mical basis of drug action as they were over half a century ago. Sections in that 1937 volume such as "Curves relating exposure to drugs with biological effects" and "Implications of monomolecular theory" show Clark's keen insight into the essential questions that required answers to move pharmaƯ cology forward. With the advent of molecular biological cloning of human receptors has come a transformation of receptor pharmacology. Thus the expression of human receptors into surrogate host cells helped unlock secrets of receptor mechanisms and stimulus-transduction pathways. To a large extent, this elimƯ inates the leap of faith required to apply receptor activity of drugs tested on animal receptor systems to the human therapeutic arena. However, a new leap of faith concerning the veracity of the effects found in recombinant systems with respect to natural ones is now required.

### Heavy Tailed Functional Time Series by Thomas Meinguet Book Summary:

The goal of this thesis is to treat the temporal tail dependence and the cross-sectional tail dependence of heavy tailed functional time series. Functional time series are aimed at modelling spatio-temporal phenomena; for instance rain, temperature, pollution on a given geographical area, with temporally dependent observations. Heavy tails mean that the series can exhibit much higher spikes than with Gaussian distributions for instance. In such cases, second moments cannot be assumed to exist, violating the basic assumption in standard functional data analysis based on the sequence of autocovariance operators. As for random variables, regular variation provides the mathematical backbone for a coherent theory of extreme values. The main tools introduced in this thesis for a regularly varying functional time series are its tail process and its spectral process. These objects capture all the aspects of the probability distribution of extreme values jointly over time and space. The development of the tail and spectral process for heavy tailed functional time series is followed by three theoretical applications. The first application is a characterization of a variety of indices and objects describing the extremal behavior of the series: the extremal index, tail dependence coefficients, the extremogram and the point process of extremes. The second is the computation of an explicit expression of the tail and spectral processes for heavy tailed linear functional time series. The third and final application is the introduction and the study of a model for the spatio-temporal dependence for functional time series called maxima of moving maxima of continuous functions (CM3 processes), with the development of an estimation method.