Extreme Values In Random Sequences () [Mladenović] [9783031574122

Preface
Contents
1 Regularly Varying Functions
	1.1 Definition and Basic Properties
		1.1.1 Regular Variation at Infinity
		1.1.2 Regular Variation at an Arbitrary Point
		1.1.3 Regular Variation and the Cauchy Equation
		1.1.4 Uniform Convergence Theorem
		1.1.5 Fast Varying Functions
	1.2 Integral Properties of Regularly Varying Functions
		1.2.1 Karamata\'s Theorem
		1.2.2 Canonical Representation of Regularly Varying Functions
		1.2.3 Modified Form of Karamata\'s Theorems
		1.2.4 Absolutely Continuous Functions
		1.2.5 Additional Properties of Regularly Varying Functions
	1.3 Monotone Regularly Varying Functions
		1.3.1 Criteria for the Regular Variation of Monotone Functions
		1.3.2 Criteria for the Fast Variation of Monotone Functions
	1.4 Generalized Inverses
		1.4.1 Definition and Basic Properties
		1.4.2 Convergence of a Sequence of Monotone Functions
		1.4.3 Properties of the Generalized Inverses of Monotone Regularly Varying Functions
		1.4.4 Khinchin\'s Theorem
	1.5 P-Variation and G-Variation
		1.5.1 P-Varying Functions
		1.5.2 Relation Between P-Variation and Slow Variation
		1.5.3 G-Varying Functions
		1.5.4 Relations Between G-Variation and P-Variation
		1.5.5 Relation Between G-Variation and Fast Variation
		1.5.6 Mejzler–de Haan Theorem
	1.6 Regularly Varying Random Variables
		1.6.1 Definition and Basic Properties
		1.6.2 Sum of Regularly Varying Random Variables
		1.6.3 Product of Random Variables
		1.6.4 Regular Variation of Random Vectors
	Exercises and Supplements
2 Basic Results of Extreme Value Theory
	2.1 Extreme Value Distributions
		2.1.1 Introduction
		2.1.2 The α-Parametrization and the γ-Parametrization
		2.1.3 Auxiliary Results
		2.1.4 Maximum Stable Distributions
	2.2 Extremal Type Theorem
	2.3 Domains of Attraction of EV Distributions
		2.3.1 Domain of Attraction of the Fréchet Distribution
		2.3.2 Domain of Attraction of the Weibull Distribution
		2.3.3 Domain of Attraction of the Gumbel Distribution
		2.3.4 Distributions Out of the Domains of Attraction
		2.3.5 Mixture of Distributions and Extreme Values
	2.4 Power Normalization
	2.5 Max-semistable Distributions
	2.6 Peaks over Thresholds
		2.6.1 Excesses over a High Threshold
		2.6.2 Generalized Pareto Distributions
		2.6.3 POT Stability
		2.6.4 Domains of Attraction for Excesses and LimitDistributions
	2.7 Stationary Sequences
		2.7.1 Conditions of Weak Dependence
		2.7.2 Extremal Type Theorem for Stationary Sequences
		2.7.3 Stationary Normal Sequences
		2.7.4 Extremal Index
		2.7.5 Point Processes and Extreme Values
	Exercises and Supplements
3 Time Series and Missing Observations
	3.1 Introduction
	3.2 Gaussian Sequences
	3.3 Weakly Dependent Stationary Sequences
	3.4 A Storage Process in Discrete Time
	3.5 Random Failures or Censoring in the Observation Mechanism
		3.5.1 The Effects of Random Failures
		3.5.2 Censorships and Extreme Values
	3.6 Sub-sampled Models
	3.7 Uniform AR(1) Processes
		3.7.1 Definition and Introductory Notes
		3.7.2 Limit Theorems for Partial Maxima
		3.7.3 Proofs
		3.7.4 Condition D(un,vn)
	3.8 Linear Processes with Heavy-Tailed Innovations
		3.8.1 Preliminaries
		3.8.2 Point Process Theory Approach
		3.8.3 Linear Processes with Positive Coefficients
		3.8.4 Linear Processes with Coefficients Alternately Changing Sign
		3.8.5 On Extremes in Non-Stationary Sequences
	Exercises and Supplements
4 Combinatorial Problems and Extreme Values
	4.1 Introduction
	4.2 Random Permutations
		4.2.1 The Maximal Step
		4.2.2 Maximum of the Sum of Consecutive Terms
		4.2.3 Cycles of Random Permutations
		4.2.4 Poisson-Dirichlet Distribution and the Longest Cycles
		4.2.5 The Longest Increasing Subsequence
	4.3 The Coupon Collector\'s Problem
		4.3.1 Introduction
		4.3.2 Waiting Time for All Elements to Appear r Times
		4.3.3 The Problem of Collecting Pairs
		4.3.4 On the Rate of Convergence in Limit Theorems
		4.3.5 Point Processes Associated with the Coupon Collector\'s Problem
		4.3.6 Coupon Collector\'s Problem and Generalized Pareto Distributions
		4.3.7 Other Extensions
	4.4 The Polynomial Scheme
	4.5 Random Trees and Random Forests
		4.5.1 The Height of a Random Tree
		4.5.2 Random Forests
		4.5.3 Maximum Tree Size in a Random Forest
		4.5.4 The Height of a Random Forest
	4.6 Random Partitions of Finite Sets
		4.6.1 Partitions of Finite Sets
		4.6.2 Random Partitions
	4.7 Geometric Properties of a Sample of Random Vectors
		4.7.1 Spherically Symmetric Distributions
		4.7.2 Diameter of a Random Sample in the Unit Ball
		4.7.3 Diameter in a Sample of Vectors with Spherically Symmetric Distribution in Rn
		4.7.4 The Distance Between a Convex Set and an Interior Random Convex Hull
	Exercises and Supplements
References
Index