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This modern and comprehensive guide to long-range dependence and self-similarity starts with rigorous coverage of the basics, then moves on to cover more specialized, up-to-date topics central to current research. These topics concern, but are not limited to, physical models that give rise to long-range dependence and self-similarity; central and non-central limit theorems for long-range dependent series, and the limiting Hermite processes; fractional Brownian motion and its stochastic calculus; several celebrated decompositions of fractional Brownian motion; multidimensional models for long-range dependence and self-similarity; and maximum likelihood estimation methods for long-range dependent time series. Designed for graduate students and researchers, each chapter of the book is supplemented by numerous exercises, some designed to test the reader's understanding, while others invite the reader to consider some of the open research problems in the field today.

Self-similar processes. --- Time-series analysis. --- Gaussian processes. --- Distribution (Probability theory) --- Stochastic processes --- Analysis of time series --- Autocorrelation (Statistics) --- Harmonic analysis --- Mathematical statistics --- Probabilities --- Selfsimilar processes

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Self-similar processes are stochastic processes that are invariant in distribution under suitable time scaling, and are a subject intensively studied in the last few decades. This book presents the basic properties of these processes and focuses on the study of their variation using stochastic analysis. While self-similar processes, and especially fractional Brownian motion, have been discussed in several books, some new classes have recently emerged in the scientific literature.  Some of them are extensions of fractional Brownian motion (bifractional Brownian motion, subtractional Brownian motion, Hermite processes), while others are solutions to the partial differential equations driven by fractional noises. In this monograph the author discusses the basic properties of these new classes of  self-similar processes and their interrrelationship. At the same time a new approach (based on stochastic calculus, especially Malliavin calculus) to studying the behavior of the variations of self-similar processes has been developed over the last decade. This work surveys these recent techniques and findings on limit theorems and Malliavin calculus.

Calculus of variations. --- Calculus. --- Mathematics. --- Self-similar processes. --- Mathematics --- Physical Sciences & Mathematics --- Mathematical Statistics --- Stochastic processes. --- Random processes --- Selfsimilar processes --- Probabilities. --- Statistics. --- Probability Theory and Stochastic Processes. --- Statistics, general. --- Probabilities --- Stochastic processes --- Distribution (Probability theory. --- Statistical analysis --- Statistical data --- Statistical methods --- Statistical science --- Econometrics --- Distribution functions --- Frequency distribution --- Characteristic functions --- Statistics . --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk

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The modeling of stochastic dependence is fundamental for understanding random systems evolving in time. When measured through linear correlation, many of these systems exhibit a slow correlation decay--a phenomenon often referred to as long-memory or long-range dependence. An example of this is the absolute returns of equity data in finance. Selfsimilar stochastic processes (particularly fractional Brownian motion) have long been postulated as a means to model this behavior, and the concept of selfsimilarity for a stochastic process is now proving to be extraordinarily useful. Selfsimilarity translates into the equality in distribution between the process under a linear time change and the same process properly scaled in space, a simple scaling property that yields a remarkably rich theory with far-flung applications. After a short historical overview, this book describes the current state of knowledge about selfsimilar processes and their applications. Concepts, definitions and basic properties are emphasized, giving the reader a road map of the realm of selfsimilarity that allows for further exploration. Such topics as noncentral limit theory, long-range dependence, and operator selfsimilarity are covered alongside statistical estimation, simulation, sample path properties, and stochastic differential equations driven by selfsimilar processes. Numerous references point the reader to current applications. Though the text uses the mathematical language of the theory of stochastic processes, researchers and end-users from such diverse fields as mathematics, physics, biology, telecommunications, finance, econometrics, and environmental science will find it an ideal entry point for studying the already extensive theory and applications of selfsimilarity.

Self-similar processes. --- Distribution (Probability theory) --- Processus autosimilaires --- Distribution (Théorie des probabilités) --- 519.218 --- Self-similar processes --- 519.24 --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Selfsimilar processes --- Stochastic processes --- Special stochastic processes --- 519.218 Special stochastic processes --- Distribution (Théorie des probabilités) --- Almost surely. --- Approximation. --- Asymptotic analysis. --- Autocorrelation. --- Autoregressive conditional heteroskedasticity. --- Autoregressive–moving-average model. --- Availability. --- Benoit Mandelbrot. --- Brownian motion. --- Central limit theorem. --- Change of variables. --- Computational problem. --- Confidence interval. --- Correlogram. --- Covariance matrix. --- Data analysis. --- Data set. --- Determination. --- Fixed point (mathematics). --- Foreign exchange market. --- Fractional Brownian motion. --- Function (mathematics). --- Gaussian process. --- Heavy-tailed distribution. --- Heuristic method. --- High frequency. --- Inference. --- Infimum and supremum. --- Instance (computer science). --- Internet traffic. --- Joint probability distribution. --- Likelihood function. --- Limit (mathematics). --- Linear regression. --- Log–log plot. --- Marginal distribution. --- Mathematica. --- Mathematical finance. --- Mathematics. --- Methodology. --- Mixture model. --- Model selection. --- Normal distribution. --- Parametric model. --- Power law. --- Probability theory. --- Publication. --- Random variable. --- Regime. --- Renormalization. --- Result. --- Riemann sum. --- Self-similar process. --- Self-similarity. --- Simulation. --- Smoothness. --- Spectral density. --- Square root. --- Stable distribution. --- Stable process. --- Stationary process. --- Stationary sequence. --- Statistical inference. --- Statistical physics. --- Statistics. --- Stochastic calculus. --- Stochastic process. --- Technology. --- Telecommunication. --- Textbook. --- Theorem. --- Time series. --- Variance. --- Wavelet. --- Website.