Extremes and Related Properties of Random Sequences and Processes

 
9781461254515: Extremes and Related Properties of Random Sequences and Processes

Classical Extreme Value Theory-the asymptotic distributional theory for maxima of independent, identically distributed random variables-may be regarded as roughly half a century old, even though its roots reach further back into mathematical antiquity. During this period of time it has found significant application-exemplified best perhaps by the book Statistics of Extremes by E. J. Gumbel-as well as a rather complete theoretical development. More recently, beginning with the work of G. S. Watson, S. M. Berman, R. M. Loynes, and H. Cramer, there has been a developing interest in the extension of the theory to include, first, dependent sequences and then continuous parameter stationary processes. The early activity proceeded in two directions-the extension of general theory to certain dependent sequences (e.g., Watson and Loynes), and the beginning of a detailed theory for stationary sequences (Berman) and continuous parameter processes (Cramer) in the normal case. In recent years both lines of development have been actively pursued.

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Contenuti:

I Classical Theory of Extremes.- 1 Asymptotic Distributions of Extremes.- 1.1. Introduction and Framework.- 1.2. Inverse Functions and Khintchine’s Convergence Theorem.- 1.3. Max-Stable Distributions.- 1.4. Extremal Types Theorem.- 1.5. Convergence of PMn < un.- 1.6. General Theory of Domains of Attraction.- 1.7. Examples.- 1.8. Minima.- 2 Exceedances of Levels and kth Largest Maxima.- 2.1. Poisson Properties of Exceedances.- 2.2. Asymptotic Distribution of kth Largest Values.- 2.3. Joint Asymptotic Distribution of the Largest Maxima.- 2.4. Rate of Convergence.- 2.5. Increasing Ranks.- 2.6. Central Ranks.- 2.7. Intermediate Ranks.- 2.1. Part II Extremal Properties of Dependent Sequences.- 3 Maxima of Stationary Sequences.- 3.1. Dependence Restrictions for Stationary Sequences.- 3.2. Distributional Mixing.- 3.3. Extremal Types Theorem for Stationary Sequences.- 3.4. Convergence of PMn ? un} Under Dependence.- 3.5. Associated Independent Sequences and Domains of Attraction.- 3.6. Maxima Over Arbitrary Intervals.- 3.7. On the Roles of the Conditions D(un), D’(un).- 3.8. Maxima of Moving Averages of Stable Variables.- 4 Normal Sequences.- 4.1. Stationary Normal Sequences and Covariance Conditions.- 4.2. Normal Comparison Lemma.- 4.3. Extremal Theory for Normal Sequences?Direct Approach.- 4.4. The Conditions D(un), D’(un) for Normal Sequences.- 4.5. Weaker Dependence Assumptions.- 4.6. Rate of Convergence.- 5 Convergence of the Point Process of Exceedances, and the Distribution of kth Largest Maxima.- 5.1. Point Processes of Exceedances.- 5.2. Poisson Convergence of High-Level Exceedances.- 5.3. Asymptotic Distribution of kth Largest Values.- 5.4. Independence of Maxima in Disjoint Intervals.- 5.5. Exceedances of Multiple Levels.- 5.6. Joint Asymptotic Distribution of the Largest Maxima.- 5.7. Complete Poisson Convergence.- 5.8. Record Times and Extremal Processes.- 6 Nonstationary, and Strongly Dependent Normal Sequences.- 6.1. Nonstationary Normal Sequences.- 6.2. Asymptotic Distribution of the Maximum.- 6.3. Convergence of 12 Under Weakest Conditions on uni.- 6.4. Stationary Normal Sequences with Strong Dependence.- 6.5. Limits for Exceedances and Maxima when rn log n ? ? < ?.- 6.6. Distribution of the Maximum when rn log n ?; ?.- 6.1. Part III Extreme Values in Continuous Time.- 7 Basic Properties of Extremes and Level Crossings.- 7.1. Framework.- 7.2. Level Crossings and Their Basic Properties.- 7.3. Crossings by Normal Processes.- 7.4. Maxima of Normal Processes.- 7.5. Marked Crossings.- 7.6. Local Maxima.- 8 Maxima of Mean Square Differentiable Normal Processes.- 8.1. Conditions.- 8.2. Double Exponential Distribution of the Maximum.- 9 Point Processes of Upcrossings and Local Maxima.- 9.1. Poisson Convergence of Upcrossings.- 9.2. Full Independence of Maxima in Disjoint Intervals.- 9.3. Upcrossings of Several Adjacent Levels.- 9.4. Location of Maxima.- 9.5. Height and Location of Local Maxima.- 9.6. Maxima Under More General Conditions.- 10 Sample Path Properties at Upcrossings.- 10.1. Marked Upcrossings.- 10.2. Empirical Distributions of the Marks at Upcrossings.- 10.3. The Slepian Model Process.- 10.4. Excursions Above a High Level.- 11 Maxima and Minima and Extremal Theory for Dependent Processes.- 11.1. Maxima and Minima.- 11.2. Extreme Values and Crossings for Dependent Processes.- 12 Maxima and Crossings of Nondifferentiable Normal Processes.- 12.1. Introduction and Overview of the Main Result.- 12.2. Maxima Over Finite Intervals.- 12.3. Maxima Over Increasing Intervals.- 12.4. Asymptotic Properties of ?-upcrossings.- 12.5. Weaker Conditions at Infinity.- 13 Extremes of Continuous Parameter Stationary Processes.- 13.1. The Extremal Types Theorem.- 13.2. Convergence of P{M(T) ?}uT.- 13.3. Associated Sequence of Independent Variables.- 13.4. Stationary Normal Processes.- 13.5. Processes with Finite Upcrossing Intensities.- 13.6. Poisson Convergence of Upcrossings.- 13.7. Interpretation of the Function ?(u).- Applications of Extreme Value Theory.- 14 Extreme Value Theory and Strength of Materials.- 14.1. Characterizations of the Extreme Value Distributions.- 14.2. Size Effects in Extreme Value Distributions.- 15 Application of Extremes and Crossings Under Dependence.- 15.1. Extremes in Discrete and Continuous Time.- 15.2. Poisson Exceedances and Exponential Waiting Times.- 15.3. Domains of Attraction and Extremes from Mixed Distributions.- 15.4. Extrapolation of Extremes Over an Extended Period of Time.- 15.5. Local Extremes?Application to Random Waves.- Appendix Some Basic Concepts of Point Process Theory.- List of Special Symbols.

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Leadbetter, M. R.; Lindgren, Georg; Rootzen, Holger
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Descrizione libro Springer-Verlag New York Inc., United States, 2012. Paperback. Condizione libro: New. 229 x 155 mm. Language: English . Brand New Book ***** Print on Demand *****.Classical Extreme Value Theory-the asymptotic distributional theory for maxima of independent, identically distributed random variables-may be regarded as roughly half a century old, even though its roots reach further back into mathematical antiquity. During this period of time it has found significant application-exemplified best perhaps by the book Statistics of Extremes by E. J. Gumbel-as well as a rather complete theoretical development. More recently, beginning with the work of G. S. Watson, S. M. Berman, R. M. Loynes, and H. Cramer, there has been a developing interest in the extension of the theory to include, first, dependent sequences and then continuous parameter stationary processes. The early activity proceeded in two directions-the extension of general theory to certain dependent sequences (e.g., Watson and Loynes), and the beginning of a detailed theory for stationary sequences (Berman) and continuous parameter processes (Cramer) in the normal case. In recent years both lines of development have been actively pursued. Softcover reprint of the original 1st ed. 1983. Codice libro della libreria AAV9781461254515

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Leadbetter, M. R.; Lindgren, Georg; Rootzen, Holger
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Descrizione libro Springer-Verlag New York Inc., United States, 2012. Paperback. Condizione libro: New. 229 x 155 mm. Language: English . Brand New Book ***** Print on Demand *****. Classical Extreme Value Theory-the asymptotic distributional theory for maxima of independent, identically distributed random variables-may be regarded as roughly half a century old, even though its roots reach further back into mathematical antiquity. During this period of time it has found significant application-exemplified best perhaps by the book Statistics of Extremes by E. J. Gumbel-as well as a rather complete theoretical development. More recently, beginning with the work of G. S. Watson, S. M. Berman, R. M. Loynes, and H. Cramer, there has been a developing interest in the extension of the theory to include, first, dependent sequences and then continuous parameter stationary processes. The early activity proceeded in two directions-the extension of general theory to certain dependent sequences (e.g., Watson and Loynes), and the beginning of a detailed theory for stationary sequences (Berman) and continuous parameter processes (Cramer) in the normal case. In recent years both lines of development have been actively pursued. Softcover reprint of the original 1st ed. 1983. Codice libro della libreria AAV9781461254515

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Descrizione libro Springer Nov 2011, 2011. Taschenbuch. Condizione libro: Neu. 235x155x18 mm. This item is printed on demand - Print on Demand Neuware - Classical Extreme Value Theory-the asymptotic distributional theory for maxima of independent, identically distributed random variables-may be regarded as roughly half a century old, even though its roots reach further back into mathematical antiquity. During this period of time it has found significant application-exemplified best perhaps by the book Statistics of Extremes by E. J. Gumbel-as well as a rather complete theoretical development. More recently, beginning with the work of G. S. Watson, S. M. Berman, R. M. Loynes, and H. Cramer, there has been a developing interest in the extension of the theory to include, first, dependent sequences and then continuous parameter stationary processes. The early activity proceeded in two directions-the extension of general theory to certain dependent sequences (e.g., Watson and Loynes), and the beginning of a detailed theory for stationary sequences (Berman) and continuous parameter processes (Cramer) in the normal case. In recent years both lines of development have been actively pursued. 352 pp. Englisch. Codice libro della libreria 9781461254515

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Descrizione libro Springer, 2011. Paperback. Condizione libro: New. book. Codice libro della libreria 1461254515

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