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.

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.