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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.