Contributions to Probability
A Collection of Papers Dedicated to Eugene Lukacs
- 1st Edition - May 10, 2014
- Editors: J. Gani, V. K. Rohatgi
- Language: English
- Paperback ISBN:9 7 8 - 1 - 4 8 3 2 - 3 9 2 0 - 0
- eBook ISBN:9 7 8 - 1 - 4 8 3 2 - 6 2 5 6 - 7
Contributions to Probability: A Collection of Papers Dedicated to Eugene Lukacs is a collection of papers that reflect Professor Eugene Lukacs’ broad range of research interests.… Read more
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Request a sales quoteContributions to Probability: A Collection of Papers Dedicated to Eugene Lukacs is a collection of papers that reflect Professor Eugene Lukacs’ broad range of research interests. This text celebrates the 75th birthday of Eugene Lukacs, mathematician, teacher, and research worker in probability and mathematical statistics. This book is organized into two parts encompassing 23 chapters. Part I consists of papers in probability theory, limit theorems, and stochastic processes. This part also deals with the continuation and arithmetic of distribution functions, the arc sine law, Fourier transform methods, and nondifferentiality of the Wiener sheet. Part II includes papers in information and statistical theories. This book will prove useful to statisticians, mathematicians, and advance mathematics students.
List of Contributors
Preface
Eugene Lukacs
Part I. Probability
Probability Theory
Lagrange's Theorem and Thin Subsequences of Squares
References
A Kind of Random (= Stochastic) Integral
0. Random Functions
1. Measurable Random Functions
2. Random ( = Stochastic) Measure
3. Random Integral
References
Continuation of Distribution Functions
1. Introduction
2. The Normal Distribution Function
3. Analytic Distribution Functions
4. Continuation Theorems for Special Classes of i.d. d.f.
5. Uniqueness of Symmetric Distribution Functions
References
The Arc Sine Law of Paul Lévy
1. Introduction
2. The Coin Tossing Game
3. Paul Lévy's Arc Sine Law
4. An Alternative Approach
5. Exact Distributions
6. Notes
7. Lévy's Heuristic Method
8. The Case of Independent Sequences
9. The Case of Dependent Sequences
References
Limit Theorems
General Limit Theorems for Products with Applications to Convolution Products of Measures
1. Introduction
2. Some Fundamental Identities and Inequalities
3. Convergence of Powers in Seminorms to Infinitely Divisible Elements
4. Convergence of Products in Seminorms
References
Stable Limit Law and Weak Law of Large Numbers for Hilbert Space with "Large-O" Rates
1. Introduction
2. Notations and Preliminaries
3. Two General Large-O Approximation Theorems
4. Stable Limit Law on H with Rates
5. The Central Limit Theorem on H
6. The Weak Law of Large Numbers
7. Limit Theorems for Random Vectors in ℝm
References
The Arithmetic of Distribution Functions
1. Introduction
2. The Class L
3. Infinitely Divisible Characteristic Functions with Absolutely Continuous Spectral Functions
4. Products of Poisson-Type Characteristic Functions
5. Independent Sets
6. Infinitely Divisible Characteristic Functions with Continuous Spectral Functions
7. Indecomposable Laws
8. Indecomposable Factors
9. α Decompositions
References
On the Tails of a Class of Infinitely Divisible Distributions
1. Introduction
2. Some Lemmas
3. Proof of Theorem 1.2
4. On Another Theorem of Elliott and Erdös
References
Fourier Transform Methods in the Study of Limit Theorems in a Hilbert Space
1. Introduction
2. Preliminaries
3. Infinitely Divisible Probability Measures
4. The General Central Limit Problem in a Hilbert Space
5. Self-Decomposable and Stable Measures in ℋ
6. Semi-Stable Measures in ℋ
References
Polynomials in Gaussian Variables and Infinite Divisibility?
References
On the Nondifferentiability of the Wiener Sheet
1. Introduction
2. The Nondifferentiability of the Wiener Sheet in the Direction of the X Axis
3. The Wiener Sheet Is Nowhere Differentiable in Any Direction
References
The Degree of Vertices on a Randomly Growing Tree
1. Introduction
2. The State-Homogeneous Case
3. A Simple Practical Example
References
Uniform Convergence of Random Trigonometric Series and Sample Continuity of Weakly Stationary Processes
1. Introduction
2. Sequences of Random Variables Satisfying the Condition Mr
3. Uniform Convergence of Random Trigonometric Series
4. Weakly Stationary Process of Class Mr: Approximate Fourier Series
5. Sample Continuity of a Weakly Stationary Process of Mr
References
Stochastic Equations Driven by Random Measures and Semimartingales
Introduction
Notations
1. Some Preliminaries on Random Measures
2. Stochastic Equations Driven by White Random Measures
3. Equations Driven by Random Measures and Semimartingales
References
Part II. Applications of Probability
Information Theory
Derivations and Information Functions (A Tale of Two Surprises and a Half)
1. Introduction
2. General and Regular Information Functions
3. Nonnegative Information Functions
4. Solutions of (12)
References
Entropy of the Sum of Independent Bernoulli Random Variables and of the Multinomial Distribution
1. Introduction
2. Entropy of Sum of Independent Bernoulli Random Variables
3. Entropy of the Multinomial Distribution
References
On the Concept and Measure of Information Contained in an Observation
Introduction
1. Derivation of a Measure for the Information Contained in an Observation
2. The Average Information
3. Fisher Information, as Information Density
4. Connection with the Variance Bound of the Cramér-Fréchet-Rao Inequality
5. The Bayesian Approach
6. Can the Measure of Information be a Distance in the Parameter Space?
References
Statistical Theory
The Empirical Characteristic Process When Parameters Are Estimated
1. Introduction
2. Notation
3. Assumptions
4. The Integrated Squared Error Estimator
5. The Integrated Error Estimator
6. The Parameter Estimated Empirical Characteristic Process with the Integrated Squared Error Estimator
7. The Parameter Estimated Empirical Characteristic Process with the Integrated Error Estimator
8. Remarks
References
Identifiability
1. Introduction
2. Identification in Errors in the Variables Models
3. Identifiability of ARMA Systems
References
On a Multivariate Extension of the Behrens - Fisher Law
References
Constant Regression of Quadratic Statistics on the Sum of Random Variables Defined on a Markov Chain
1. Introduction
2. Preliminaries
3. On the Constant Regression of Quadratic Forms on the Sum of Random Variables Defined on a Markov Chain
4. The Characterization of the Normal Law by the Constant Regression of Quadratic Statistics on the Sum of Random Variables Defined on a Markov Chain
References
Invariance Principles for Rank Statistics for Testing Independence
1. Introduction
2. Assumptions and Some Preliminary Lemmas
3. Order of Magnitude of the Remainder Term
4. Invariance Principles
References
Integral Transformations of Distributions and Estimates of Parameters of Multidimensional Spherically Symmetric Stable Laws
1. Introduction
2. Some Properties of Spherically Symmetric Distributions
3. Integral Transformations of Distributions
4. The Estimation of Parameters of Spherically Symmetric Stable Laws
5. Comments
References
Eugene Lukacs: Bibliography
- No. of pages: 334
- Language: English
- Edition: 1
- Published: May 10, 2014
- Imprint: Academic Press
- Paperback ISBN: 9781483239200
- eBook ISBN: 9781483262567
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