Fractional Brownian Motions, Fractional Noises and Applications

Meyn & Tweedie is back! The bible on Markov chains in general state spaces has been brought up to date to reflect developments in the field since 1996 - many of them sparked by publication of the first edition. The pursuit of more efficient simulation algorithms for complex Markovian models, or algorithms for computation of optimal policies for controlled Markov models, has opened new directions for research on Markov chains. As a result, new applications have emerged across a wide range of topics including optimisation, statistics, and economics. New commentary and an epilogue by Sean Meyn summarise recent developments and references have been fully updated. This second edition reflects the same discipline and style that marked out the original and helped it to become a classic: proofs are rigorous and concise, the range of applications is broad and knowledgeable, and key ideas are accessible to practitioners with limited mathematical background.

Markov Chains and Stochastic Stability

Couplings and changes of variables.- Three examples of coupling techniques.- The founding fathers of optimal transport.- Qualitative description of optimal transport.- Basic properties.- Cyclical monotonicity and Kantorovich duality.- The Wasserstein distances.- Displacement interpolation.- The Monge-Mather shortening principle.- Solution of the Monge problem I: global approach.- Solution of the Monge problem II: Local approach.- The Jacobian equation.- Smoothness.- Qualitative picture.- Optimal transport and Riemannian geometry.- Ricci curvature.- Otto calculus.- Displacement convexity I.- Displacement convexity II.- Volume control.- Density control and local regularity.- Infinitesimal displacement convexity.- Isoperimetric-type inequalities.- Concentration inequalities.- Gradient flows I.- Gradient flows II: Qualitative properties.- Gradient flows III: Functional inequalities.- Synthetic treatment of Ricci curvature.- Analytic and synthetic points of view.- Convergence of metric-measure spaces.- Stability of optimal transport.- Weak Ricci curvature bounds I: Definition and Stability.- Weak Ricci curvature bounds II: Geometric and analytic properties.

Optimal Transport: Old and New

This book is an introduction to probability theory covering laws of large numbers, central limit theorems, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion. It is a comprehensive treatment concentrating on the results that are the most useful for applications. Its philosophy is that the best way to learn probability is to see it in action, so there are 200 examples and 450 problems.

Probability: Theory and Examples

Thirty-seven programs that summarize data with histograms and other graphics, calculate measures of spatial continuity, provide smooth least-squares-type maps, and perform stochastic spatial simulation.

GSLIB: Geostatistical Software Library and User's Guide

This note discusses some aspects of the estimation of the density function of a univariate probability distribution. All estimates of the density function satisfying relatively mild conditions are shown to be biased. The asymptotic mean square error of a particular class of estimates is evaluated.

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Remarks on Some Nonparametric Estimates of a Density Function

/pdf/remarks-on-a-multivariate-transformation-34okbvxuxl.pdf

Remarks on a Multivariate Transformation

* The research for this paper was supported by the United States Air Force under Contract No. AF18(600-685) monitored by the Office of Scientific Research. 1 A discussion of the problem along with the necessary references will be found in Harry Pollard, "The Harmonic Analysis of Bounded Functions," Duke Math. / . , 20, 499-512, 1953. 2 See ibid. 3 See S. Bochner, Fouriersche Integrate (Leipzig, 1932), p. 33. 4 The Levitan polynomial. See N.I. Achieser, Approximationstheorie (Berlin, 1953), p. 146.

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A central limit theorem and a strong mixing condition.

Stationary Stochastic Processes and Their Representations: 1.0 Introduction 1.1 What is a stochastic process? 1.2 Continuity in the mean 1.3 Stochastic set functions of orthogonal increments 1.4 Orthogonal representations of stochastic processes 1.5 Stationary processes 1.6 Representations of stationary processes 1.7 Time and ensemble averages 1.8 Vector processes 1.9 Operations on stationary processes 1.10 Harmonizable stochastic processes Statistical Questions when the Spectrum is Known (Least Squares Theory): 2.0 Introduction 2.1 Preliminaries 2.2 Prediction 2.3 Interpolation 2.4 Filtering of stationary processes 2.5 Treatment of linear hypotheses with specified spectrum Statistical Analysis of Parametric Models: 3.0 Introduction 3.1 Periodogram analysis 3.2 The variate difference method 3.3 Effect of smoothing of time series (Slutzky's theorem) 3.4 Serial correlation coefficients for normal white noise 3.5 Approximate distributions of quadratic forms 3.6 Testing autoregressive schemes and moving averages 3.7 Estimation and the asymptotic distribution of the coefficients of an autoregressive scheme 3.8 Discussion of the methods described in this chapter Estimation of the Spectrum: 4.0 Introduction 4.1 A general class of estimates 4.2 An optimum property of spectrograph estimates 4.3 A remark on the bias of spectrograph estimates 4.4 The asymptotic variance of spectrograph estimates 4.5 Another class of estimates 4.6 Special estimates of the spectral density 4.7 The mean square error of estimates 4.8 An example from statistical optics Applications: 5.0 Introduction 5.1 Derivations of spectra of random noise 5.2 Measuring noise spectra 5.3 Turbulence 5.4 Measuring turbulence spectra 5.5 Basic ideas in a statistical theory of ocean waves 5.6 Other applications Distribution of Spectral Estimates: 6.0 Introduction 6.1 Preliminary remarks 6.2 A heuristic derivation of a limit theorem 6.3 Preliminary considerations 6.4 Treatment of pure white noise 6.5 The general theorem 6.6 The normal case 6.7 Remarks on the nonnormal case 6.8 Spectral analysis with a regression present 6.9 Alternative estimates of the spectral distribution function 6.10 Alternative statistics and the corresponding limit theorems 6.11 Confidence band for the spectral density 6.12 Spectral analysis of some artificially generated time series Problems in Linear Estimation: 7.0 Preliminary discussion 7.1 Estimating regression coefficients 7.2 The regression spectrum 7.3 Asymptotic expression for the covariance matrices 7.4 Elements of the spectrum 7.5 Polynomial and trigonometric regression 7.6 More general trigonometric and polynomial regression 7.7 Some other types of regression 7.8 Detection of signals in noise 7.9 Confidence intervals and tests Assorted Problems: 8.0 Introduction 8.1 Prediction when the conjectured spectrum is not the true one 8.2 Uniform convergence of the estimated spectral density to the true spectral density 8.3 The asymptotic distribution of an integral of a spectrograph estimate 8.4 The mean square error of prediction when the spectrum is estimated 8.5 Other types of estimates of the spectrum 8.6 The zeros and maxima of stationary stochastic processes 8.7 Prefiltering of a time series 8.8 Comments on tests of normality Problems Appendix on complex variable theory Bibliography Index.

Statistical analysis of stationary time series

I Stationary Processes.- 1. General Discussion.- 2. Positive Definite Functions.- 3. Fourier Representation of a Weakly Stationary Process.- Problems.- Notes.- II Prediction and Moments.- 1. Prediction.- 2. Moments and Cumulants.- 3. Autoregressive and Moving Average Processes.- 4. Non-Gaussian Linear Processes.- 5. The Kalman-Bucy Filter.- Problems.- Notes.- III Quadratic Forms, Limit Theorems and Mixing Conditions.- 1. Introduction.- 2. Quadratic Forms.- 3. A Limit Theorem.- 4. Summability of Cumulants.- 5. Long-range Dependence.- 6. Strong Mixing and Random Fields.- Problems.- Notes.- IV Estimation of Parameters of Finite Parameter Models.- 1. Maximum Likelihood Estimates.- 2. The Newton-Raphson Procedure and Gaussian ARMA Schemes.- 3. Asymptotic Properties of Some Finite Parameter Estimates.- 4. Sample Computations Using Monte Carlo Simulation.- 5. Estimating the Order of a Model.- 6. Finite Parameter Stationary Random Fields.- Problems.- V Spectral Density Estimates.- 1. The Periodogram.- 2. Bias and Variance of Spectral Density Estimates.- 3. Asymptotic Distribution of Spectral Density Estimates.- 4. Prewhitening and Tapering.- 5. Spectral Density Estimates Using Blocks.- 6. A Lower Bound for the Precision of Spectral Density Estimates.- 7. Turbulence and the Kolmogorov Spectrum.- 8. Spectral Density Estimates for Random Fields.- Problems.- Notes.- VI Cumulant Spectral Estimates.- 1. Introduction.- 2. The Discrete Fourier Transform and Fast Fourier Transform.- 3. Vector-Valued Processes.- 4. Smoothed Periodograms.- 5. Aliasing and Discretely Sampled Time Series.- Notes.- VII Density and Regression Estimates.- 1. Introduction. The Case of Independent Observations.- 2. Density and Regression Estimates for Stationary Sequences.- Notes.- VIII Non-Gaussian Linear Processes.- 1. Estimates of Phase, Coefficients, and Deconvolution for Non-Gaussian.- Linear Processes.- 2. Random Fields.- 3. Non-Gaussian Linear Random Fields.- Notes.- 1. Monotone Functions and Measures.- 2. Hilbert Space.- 3. Banach Space.- 4. Banach Algebras and Homomorphisms.- Postscript.- Author Index.

Murray Rosenblatt

Papers

Remarks on Some Nonparametric Estimates of a Density Function

Remarks on a Multivariate Transformation

A central limit theorem and a strong mixing condition.

Statistical analysis of stationary time series

Stationary sequences and random fields