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Table of Contents
Preface; Contents; 1 Stationary Processes; 1.1 Stationarity and Invariance; 1.2 Stationary Processes with a Finite Variance; 1.3 Measurability and Continuity in Probability; 1.4 Linear Processes; 1.5 Comments on Chapter 1; 1.6 Exercises to Chapter 1; 2 Elements of Ergodic Theory of Stationary Processes and Strong Mixing; 2.1 Basic Definitions and Ergodicity; 2.2 Mixing and Weak Mixing; 2.3 Strong Mixing; 2.4 Conservative and Dissipative Maps; 2.5 Comments on Chapter 2; 2.6 Exercises to Chapter 2; 3 Infinitely Divisible Processes.
3.1 Infinitely Divisible Random Variables, Vectors, and Processes3.2 Infinitely Divisible Random Measures; 3.3 Infinitely Divisible Processes as Stochastic Integrals; 3.4 Series Representations; 3.5 Examples of Infinitely Divisible Self-Similar Processes; 3.6 Stationary Infinitely Divisible Processes; 3.7 Comments on Chapter 3; 3.8 Exercises to Chapter 3; 4 Heavy Tails; 4.1 What Are Heavy Tails? Subexponentiality; 4.2 Regularly Varying Random Variables; 4.3 Multivariate Regularly Varying Tails; 4.4 Heavy Tails and Convergence of Random Measures; 4.5 Comments on Chapter 4.
4.6 Exercises to Chapter 45 Introduction to Long-Range Dependence; 5.1 The Hurst Phenomenon; 5.2 The Joseph Effect and Nonstationarity; 5.3 Long Memory, Mixing, and Strong Mixing; 5.4 Comments on Chapter 5; 5.5 Exercises to Chapter 5; 6 Second-Order Theory of Long-Range Dependence; 6.1 Time-Domain Approaches; 6.2 Spectral Domain Approaches; 6.3 Pointwise Transformations of Gaussian Processes; 6.4 Comments on Chapter 6; 6.5 Exercises to Chapter 6; 7 Fractionally Differenced and Fractionally Integrated Processes; 7.1 Fractional Integration and Long Memory.
7.2 Fractional Integration of Second-Order Processes7.3 Fractional Integration of Processes with Infinite Variance; 7.4 Comments on Chapter 7; 7.5 Exercises to Chapter 7; 8 Self-Similar Processes; 8.1 Self-Similarity, Stationarity, and Lamperti's Theorem; 8.2 General Properties of Self-Similar Processes; 8.3 SSSI Processes with Finite Variance; 8.4 SSSI Processes Without a Finite Variance; 8.5 What Is in the Hurst Exponent? Ergodicity and Mixing; 8.6 Comments on Chapter 8; 8.7 Exercises to Chapter 8; 9 Long-Range Dependence as a Phase Transition; 9.1 Why Phase Transitions?
9.2 Phase Transitions in Partial Sums9.3 Partial Sums of Finite-Variance Linear Processes; 9.4 Partial Sums of Finite-Variance Infinitely Divisible Processes; 9.5 Partial Sums of Infinite-Variance Linear Processes; 9.6 Partial Sums of Infinite-Variance Infinitely Divisible Processes; 9.7 Phase Transitions in Partial Maxima; 9.8 Partial Maxima of Stationary Stable Processes; 9.9 Comments on Chapter 9; 9.10 Exercises to Chapter 9; 10 Appendix; 10.1 Topological Groups; 10.2 Weak and Vague Convergence; 10.3 Signed Measures; 10.4 Occupation Measures and Local Times.
3.1 Infinitely Divisible Random Variables, Vectors, and Processes3.2 Infinitely Divisible Random Measures; 3.3 Infinitely Divisible Processes as Stochastic Integrals; 3.4 Series Representations; 3.5 Examples of Infinitely Divisible Self-Similar Processes; 3.6 Stationary Infinitely Divisible Processes; 3.7 Comments on Chapter 3; 3.8 Exercises to Chapter 3; 4 Heavy Tails; 4.1 What Are Heavy Tails? Subexponentiality; 4.2 Regularly Varying Random Variables; 4.3 Multivariate Regularly Varying Tails; 4.4 Heavy Tails and Convergence of Random Measures; 4.5 Comments on Chapter 4.
4.6 Exercises to Chapter 45 Introduction to Long-Range Dependence; 5.1 The Hurst Phenomenon; 5.2 The Joseph Effect and Nonstationarity; 5.3 Long Memory, Mixing, and Strong Mixing; 5.4 Comments on Chapter 5; 5.5 Exercises to Chapter 5; 6 Second-Order Theory of Long-Range Dependence; 6.1 Time-Domain Approaches; 6.2 Spectral Domain Approaches; 6.3 Pointwise Transformations of Gaussian Processes; 6.4 Comments on Chapter 6; 6.5 Exercises to Chapter 6; 7 Fractionally Differenced and Fractionally Integrated Processes; 7.1 Fractional Integration and Long Memory.
7.2 Fractional Integration of Second-Order Processes7.3 Fractional Integration of Processes with Infinite Variance; 7.4 Comments on Chapter 7; 7.5 Exercises to Chapter 7; 8 Self-Similar Processes; 8.1 Self-Similarity, Stationarity, and Lamperti's Theorem; 8.2 General Properties of Self-Similar Processes; 8.3 SSSI Processes with Finite Variance; 8.4 SSSI Processes Without a Finite Variance; 8.5 What Is in the Hurst Exponent? Ergodicity and Mixing; 8.6 Comments on Chapter 8; 8.7 Exercises to Chapter 8; 9 Long-Range Dependence as a Phase Transition; 9.1 Why Phase Transitions?
9.2 Phase Transitions in Partial Sums9.3 Partial Sums of Finite-Variance Linear Processes; 9.4 Partial Sums of Finite-Variance Infinitely Divisible Processes; 9.5 Partial Sums of Infinite-Variance Linear Processes; 9.6 Partial Sums of Infinite-Variance Infinitely Divisible Processes; 9.7 Phase Transitions in Partial Maxima; 9.8 Partial Maxima of Stationary Stable Processes; 9.9 Comments on Chapter 9; 9.10 Exercises to Chapter 9; 10 Appendix; 10.1 Topological Groups; 10.2 Weak and Vague Convergence; 10.3 Signed Measures; 10.4 Occupation Measures and Local Times.