CHAPTER I THE EXPONENTIAL AND THE UNIFORM DENSITES 1. Introduction 2. Densities. Convolutions 3. The Exponential Density 4. Waiting Time Paradoxes. The Poisson Process 5. The Persistence of Bad Luck 6. Waiting Times and Order Statistics 7. The Uniform Distribution 8. Random Splittings 9. Convolutions and Covering Theorems 10. Random Directions 11. The Use of Lebesgue Measure 12. Empirical Distributions 13. Problems for Solution CHAPTER II SPECIAL DENSITIES. RANDOMIZATION 1. Notations and Conventions 2. Gamma Distributions 3. Related Distributions of Statistics 4. Some Common Densities 5. Randomization and Mixtures 6. Discrete Distributions 7. Bessel Functions and Random Walks 8. Distributions on a Circle 9. Problems for Solution CHAFFER III DENSITIES IN HIGHER DIMENSIONS. NORMAL DENSTTES AND PROCESSES 1. Densities 2. Conditional Distributions 3. Return to the Exponential and the Uniform Distributions 4. A Characterization of the Normal Distribution 5. Matrix Notation. The Covariance Matrix 6. Normal Densities and Distributions 7. Stationary Normal Processes 8. Markovian Normal Densities 9. Problems for Solution CHAPTER IV PROBABILITY MEASURES AND SPACES □ 1. Baire Functions 2. Interval Functions and Integrals in 3. Algebras. Measurability 4. Probability Spaces. Random Variables 5. The Extension Theorem 6. Product Spaces. Sequences of Independent Variables. 7. Null Sets. Completion CHAPTER V PROBABILITY DISTRIBUTIONS IN □ 1. Distributions and Expectations
2. Preliminaries 3. Densities 4. Convolutions …… CHAPTER VI A SURVEY OF SOME IMPORTANT DISTRIBUTIONS AND PROCESSES CHAFFER VII LAWS OF LARGE NUMBERS.APPLICATIONS IN ANALYSIS. CHAPTEK VIII THE BASIC LIMIT THEOREMS CHAPTER IX INFINITELY DIVISIBLE DISTRIBUTIONS AND SEMI.GROUPS CHAPTER X MARKOV PROCESSES AND SEMI-GROUPS CHAFFER XI RENEWAL THEORY CHAPTER XII RANDOM WALKS IN □ CHAPTER XIII LAPLACE TRANSFORMS.TAUBERIAN THEOREMS.RESOLVENTS CHAPTER XIV APPLICATIONS OF LAPLACE TRANGFORMS CHAPTER XV CHARACTERISTIC FUNCTIONS CHAPTER XVI EXPANSIONS RELATED TO THE CENTRAL LIMIT THEOREM CHAPTER XVII INFINITELY DIVISIBLE DISTRIBUTIONS. CHAPTER XVIII APPLICATIONS OF FOURIER METHODS TO RANDOM WALKS CHAPTEK XIX HARMONIC ANALYSIS ANSWERS TO PROBLEMS SOME BOOKS ON COGNATE SUBJECTS INDEX
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