Extreme Value Theory for Time Series : Models with Power-Law Tails

• Autore: Mikosch Thomas
• Pubblicazione: Cham : , : Springer, , 2024; ©2024
• Edizione: 1st ed.
• Descrizione fisica: 1 online resource (768 pages)
• Altri autori: WintenbergerOlivier
• Nota di contenuto: Intro -- Preface -- Contents -- 1 Introduction -- 1.1 From Poisson's Law to the Fréchet and Pareto Distributions: A Short Excursion into the History of Extreme Value Theory -- 1.2 Empirical Evidence of Power-Law Tails, Extremal Dependence and Clusters in Return Data -- 1.3 What Is this Book About? -- 1.4 What Is Special About this Book? -- Part I Regular Variation of Distributions and Processes -- 2 The IID Univariate Benchmark -- 2.1 Sum-Stable Distributions and Their Domains of Attraction -- 2.2 Max-Stable Distributions and Their Domains of Attraction -- 2.3 Point Processes and Order Statistics -- 2.4 Weak Convergence of Upper Order Statistics -- 2.5 Weak Convergence of the Binomial Processes to a Poisson Process -- 3 Regularly Varying Random Variables and Vectors -- 3.1 Regularly Varying Random Variables -- 3.1.1 Regularly Varying Functions -- 3.1.2 Definition of a Regularly Varying Random Variable -- 3.1.3 Operations on Regularly Varying Random Variables -- 3.1.3.1 Power Transformations and Moments -- 3.1.3.2 Closure of Regular Variation Under Convolution -- 3.1.3.3 Subexponentiality of Regularly Varying Distributions, the Convolution-Root Property, and Regularly Varying Ruin Probabilities -- 3.1.3.4 Closure of Regular Variation Under Multiplication -- 3.1.4 An Excursion to the Estimation of the Tail Index: The Hill Estimator -- 3.2 Regularly Varying Random Vectors -- 3.2.1 The Multivariate Fréchet Distribution -- 3.2.1.1 Convergence of Component-Wise Maxima to the Multivariate Fréchet Distribution -- 3.2.1.2 The Exponent Measure of the Multivariate Fréchet Distribution -- 3.2.1.3 The Max-Stability Property of the Multivariate Fréchet Distribution -- 3.2.2 Regular Variation of Random Vectors with Non-Negative Components -- 3.2.2.1 The Homogeneity Property of the Limit Measure -- 3.2.2.2 The General Multivariate Fréchet Distribution. ; 3.2.2.3 Examples with Independent or Asymptotically Independent Components -- 3.2.3 Regular Variation of Random Vectors with General Components -- 3.2.4 Equivalent Formulations of Multivariate Regular Variation in Rd -- 3.2.4.1 The Regular Variation Property does not Depend on the Choice of the Norm -- 3.2.4.2 Families of Sets that Generate the Regular Variation Property -- 3.2.4.3 The Continuous Form of Regular Variation -- 3.2.4.4 Weak Convergence of the Binomial Processes to a Poisson Process -- 3.2.4.5 Point Process Representation of a Multivariate Fréchet Distribution -- 3.2.5 Operations on Regularly Varying Vectors -- 3.2.5.1 Asymptotically Independent Random Vectors -- 3.2.5.2 Positively Homogeneous Continuous Mappings Acting on a Regularly Varying Vector -- 3.2.5.3 Linear Transformations of Regularly Varying Vectors -- 3.2.5.4 The Multivariate Breiman Result -- 4 Regularly Varying Time Series -- 4.1 Definition of a Regularly Varying Sequence -- 4.2 The Tail and Spectral Tail Processes -- 4.3 Time-Change Properties of the Spectral Tail Process -- 4.4 A Derivation of the Tail Measures from the Distribution of the Spectral Tail Process -- 5 Examples of Regularly Varying Stationary Processes -- 5.1 IID Regularly Varying Sequences -- 5.2 Regularly Varying Linear Processes -- 5.2.1 Preliminaries -- 5.2.2 Existence of a Univariate Linear Process -- 5.2.3 Existence of a Multivariate Linear Process -- 5.2.4 The Single Big Jump Principle -- 5.2.5 Regular Variation of |X| -- 5.2.6 The Regular Variation Property of a Linear Process -- 5.2.6.1 Univariate Linear Processes -- 5.2.6.2 The Vector-AR(1) Process -- 5.3 A Regularly Varying Stochastic Volatility Model -- 5.3.1 The Innovations Dominate the Volatility -- 5.3.2 The Volatility Dominates the Innovations -- 5.4 Max-Moving Average Processes -- 5.5 Max-Stable Stationary Processes with Fréchet Marginals. ; 5.5.1 A Brief Introduction to Max-Stable Processes -- 5.5.2 Regular Variation of a Stationary Max-Stable Process with Fréchet Marginals -- 5.5.3 Some Examples -- 5.5.3.1 The Brown-Resnick Process -- 5.5.3.2 Mixed Moving Maxima (M3) Processes -- 5.5.3.3 Spectrally Stationary Max-Stable Processes -- 5.6 Solution to Affine Stochastic Recurrence Equations -- 5.6.1 Existence and Stationarity -- 5.6.2 Example: The ARCH and GARCH Family -- 5.6.3 The Tails -- 5.6.3.1 Some Generalities -- 5.6.3.2 The Grincevičius-Grey Theorem -- 5.6.3.3 The Kesten-Goldie Theorem -- 5.6.3.4 The Marginal Tails of a GARCH(1,1) Process -- 5.6.3.5 The Regular Variation Property -- 5.6.3.6 Regular Variation of a GARCH(1,1) Process -- 5.6.3.7 Properties of the Spectral Tail Process -- The Sequence (sign(Θt)) -- Independence of (|Θt|)t≥0 and (|Θ-t|)t≥1 -- 5.7 A Queuing Model: The Lindley Process -- 5.8 Regular Variation of Lagged Products -- Part II Point Process Convergence and Cluster Phenomena of Time Series -- 6 Clusters of Extremes -- 6.1 Leadbetter's Approach-The Extremal Index -- 6.1.1 Leadbetter's Mixing Condition D -- 6.1.2 Leadbetter's Extremal Index -- 6.1.3 Leadbetter's Condition D' -- 6.1.4 The Extremal Index as Reciprocal of the Expected Extremal Cluster Length -- 6.2 Characterization of the Extremal Index -- 6.2.1 Approximations of the Extremal Index -- 6.2.2 The Extremal Index of a Regularly Varying Sequence -- 6.3 The Extremal Index for Particular Time Series Models -- 6.3.1 Asymptotic Independence of Block Maxima via Coupling Arguments -- 6.3.2 Causal Linear Process -- 6.3.3 A Regularly Varying Stochastic Volatility Process -- 6.3.4 Solution to a Stochastic Recurrence Equation -- 6.3.4.1 The Extremal Index of a GARCH(1,1) Process -- 6.3.4.2 A Characterization of the Extremal Index via Random Walk Theory -- 6.3.5 Max-Stable Process -- 6.3.5.1 The Case H=1/2. ; 6.3.5.2 Toward the Pickands Constant for H=1/2 -- 6.3.6 The Lindley Process -- 6.4 An Informal Discussion of Extremal Index Estimation -- 6.4.1 The Blocks Estimator -- 6.4.2 The Runs and Intervals Estimators -- 6.4.3 Northrop's Estimator -- 6.4.4 An Estimator of the Extremal Index Based on the Spectral Cluster Process -- 6.4.5 A Monte-Carlo Study of the Estimators -- 7 Point Process Convergence for Regularly Varying Sequences -- 7.1 Mixing and Anti-Clustering Conditions -- 7.2 The Spectral Cluster Process -- 7.3 Convergence to a Cluster Poisson Process -- 7.4 Properties of the Spectral Cluster Process -- 8 Applications of Point Process Convergence -- 8.1 Homogeneous Mappings -- 8.1.1 The Point Process of the Norms -- 8.1.2 The Joint Limit Distribution of Maxima and Minima -- 8.1.3 The Joint Limit Distribution of the Maximum and the Second Largest Order Statistic of the Norms -- 8.1.4 The Extremal Index Function of a Multivariate Time Series -- 8.2 The Cluster Length of the Exceedances -- 8.2.1 The Point Process with Time Stamps -- 8.2.2 The Point Process of the Exceedances -- 8.2.2.1 Convergence to a Compound Poisson Process -- 8.2.2.2 Extremal Cluster Probabilities -- 8.2.3 Order Statistics of the Norms -- 8.3 Examples of Cluster Processes -- 8.3.1 Causal Linear Process -- 8.3.1.1 The Mixing Conditions A(an) and A(an) -- 8.3.1.2 Spectral Cluster Process and Point Process Convergence -- 8.3.1.3 Some Applications -- The Joint Limit Distribution of Maxima and Minima -- 8.3.2 Solution to an Affine Stochastic Recurrence Equation -- 8.3.2.1 The Mixing Conditions A(an) and A(an) -- 8.3.2.2 Extremal Cluster Probabilities -- Part III Infinite Variance Central Limit Theory and Related Topics -- 9 Infinite-Variance Central Limit Theory -- 9.1 Preliminaries -- 9.1.1 Some Facts About Multivariate α-Stable Distributions -- 9.1.2 The Anti-Clustering Condition. ; 9.1.3 The Mixing Condition -- 9.1.4 Further Discussions on the Assumptions and Their Comparison with the Literature -- 9.2 α-Stable Central Limit Theory -- 9.3 Some Examples -- 9.3.1 α-Stable Central Limit Theorem for a Regularly Varying Stochastic Volatility Process -- 9.3.1.1 The Case When σ Is Bounded from Above and Below by Positive Constants -- 9.3.1.2 The Case of Unbounded Volatility Under Mixing -- 9.3.2 Regularly Varying Causal Linear Process -- 9.3.3 The Solution to an Affine Stochastic Recurrence Equation -- 10 Self-Normalization, Sample Autocorrelations and the Extremogram -- 10.1 Self-Normalized Sums -- 10.1.1 Preliminaries -- 10.1.2 Joint Convergence of Sums and Maxima -- 10.1.3 Ratio Limit Theory for Sums and Maxima -- 10.1.4 Limit Theory for Studentized Sums -- 10.1.5 Greenwood Statistics -- 10.1.6 Some Applications -- 10.1.6.1 The Anti-clustering and Mixing Conditions for Solutions of Affine Stochastic Recurrence Equations -- 10.1.6.2 Some Illustrations for a Causal AR(1) Process -- 10.1.6.3 Some Illustrations for the Squared Volatility of a GARCH Model -- 10.1.6.4 Consequences of the Extremal Dependence for Ratio Plots -- 10.2 The Sample Autocovariance Function -- 10.2.1 The Case 2< -- α< -- 4 -- 10.2.2 The Case 0< -- α< -- 2 -- 10.2.3 Examples -- 10.2.3.1 The Sample ACVF and ACF for a Stochastic Volatility Process -- 10.2.3.2 The Sample ACVF and ACF for the Solution to an Affine Stochastic Recurrence Equation -- 10.3 The Extremogram -- 10.3.1 The Extremogram: An Autocorrelation Function for Extreme Events -- 10.3.2 Variations on the Extremogram -- 10.3.3 Examples of Extremograms -- 10.3.3.1 Causal Linear Process -- 10.3.3.2 The Solution to an Affine Stochastic Recurrence Equation -- 10.3.3.3 Max-Stable Process with Fréchet Marginals -- 10.3.3.4 Asymptotically Independent Sequences. ; 10.3.4 The Cross-Extremogram of a Multivariate Time Series.
• Titolo autorizzato: Extreme Value Theory for Time Series
• ISBN: 9783031591563; 9783031591556
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910878979403321
• Lo trovi qui: Univ. Federico II
• Serie: Springer Series in Operations Research and Financial Engineering Series