INSEAD Library

Doriot: for 2019-2020 courses

Includes bibliographical references

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CHAPTER I T HE EXPONENTIAL . . . 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 . . . . . . AND THE UNIFORM DENSITIES . . . . . . 1 1 3 8 11 . 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 . . . . . . . . . . . . . . . 15 17 21 25 26 29 33 36 39 II SPECIAL DENSITIES . RANDOMIZATION . . . . . . . . 45 45 47 . . . . . . . . . 2. Gamma Distributions . . . . . . . . . . . *3. Related Distributions of Statistics 4. Some Common Densities . . . 5 . Randomization and Mixtures . 6. Discrete Distributions . . . 1. Notations and Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 49 53 55 * Starred sections are not required for the understanding of the sequel and should be omitted at first reading . 7 . Bessel Functions and Random Walks 8 . Distributions on a Circle . . . . 9. Problems for Solution . . . . . CHAPTER . . . . . . . . . . . . . . . . . . 58 61 64 III DENSITIES IN H IGHER D IMENSIONS . N ORMAL DENSITIES PROCESSES . . . . . . . . . . . . . AND 1. Densities . . . . . . . . . . . . . . . . 66 66 71 74 77 80 83 87 94 99 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 M EASURES . . . . . . 1. Baire Functions . . . . . . . . . . . 2. Interval Functions and Integrals in X' . . . . . 3. o-Algebras. Measurability . . . . . . . . 4. Probability Spaces. Random Variables . . . . . 5 . The Extension Theorem . . . . . . . . . 6. Product Spaces. Sequences of Independent Variables . 7 . Null Sets. Completion . . . . . . . . . AND SPACES . . 103 . 104 . 106 . 112 115 . 118 . 121 . 125 . CHAPTER v PROBABILITY DISTRIBUTIONS . 1 . Distributions and Expectations . 2. Preliminaries . . . . . . 3 . Densities . . . . . . . 4 . Convolutions . . . . . . IN Rr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 128 136 138 143 5. Symmetrization . . . . . . . . . . . . . 6. Integration by Parts . Existence of Moments . . . . 7. Chebyshev's Inequality . . . . . . . . . . 8. Further Inequalities. Convex Functions . . . . . 9. Simple Conditional Distributions . Mixtures . . . . *10. Conditional Distributions . . . . . . . . . . *11. Conditional Expectations . . . . . . . . . . 12. Problems for Solution . . . . . . . . . . CHAPTER 148 150 151 152 156 160 162 165 VI A SURVEY OF SOME IMPORTANT DISTRIBUTIONS AND PROCESSES 169 1 . Stable Distributions in R1 . . . . . . . . . 169 2 . Examples . . . . . . . . . . . . . . 173 3. Infinitely Divisible Distributions in R1. . . . . . 176 4. Processes with Independent Increments . . . . . . . . 179 * 5 . Ruin Problems in Compound Poisson Processes . . . . 182 6. Renewal Processes . . . . . . . . . . . . 184 7. Examples and Problems . . . . . . . . . . 187 8. Random Walks . . . . . . . . . . . . . 190 9. The Queuing Process . . . . . . . . . . . 194 10. Persistent and Transient Random Walks . . . . . 200 11. General Markov Chains . . . . . . . . . . 205 *12. Martingales . . . . . . . . . . . . . . 209 13. Problems for Solution . . . . . . . . . . . 215 CHAPTER VII LAWS . 219 1. Main Lemma and Notations . . . . . . . . . 219 2. Bernstein Polynomials . Absolutely Monotone Functions 222 3. Moment Problems . . . . . . . . . . . . 224 *4 . Application to Exchangeable Variables . . . . . . 228 *5. Generalized Taylor Formula and Semi-Groups . . . 230 6. Inversion Formulas for Laplace Transforms . . . . 232 OF IN L ARGE NUMBERS . APPLICATIONS ANALYSIS . *7 Laws of Large Numbers for Identically Distributed Variables . . . . . . . . . . . . . . 234 *8 Strong Laws . . . . . . . . . . . . . 237 *9 Generalization to Martingales . . . . . . . . 241 10. Problems for Solution . . . . . . . . . . . 244 . . . CHAPTER VIII THE BASIC LIMIT THEOREMS . . 1. Convergence of Measures . . 2. Special Properties . . . . 3. Distributions as Operators . 4 . The Central Limit Theorem . * 5 . Infinite Convolutions . . . 247 252 254 258 265 6 . Selection Theorems . . . . . . . . . . . . 267 *7 . Ergodic Theorems for Markov Chains . . . . . . 270 8. Regular Variation . . . . . . . . . . . . 275 *9. Asymptotic Properties of Regularly Varying Functions . 279 10. Problems for Solution . . . . . . . . . . . 284 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 CHAPTER I INFINITELY DIVISIBLE DISTRIBUTIONS AND X 1. Orientation . . . . . . . . 2 . Convolution Semi-Groups . . . 3 . Preparatory Lemmas . . . . . 4 . Finite Variances . . . . . . 5 . The Main Theorems . . . . . SEMI-GROUPS . 290 290 293 296 298 300 6. Example: Stable Semi-Groups . . . . . . . . 305 7. Triangular Arrays with Identical Distributions . . . . 308 8. Domains of Attraction . . . . . . . . . . 312 9 Variable Distributions . The Three-Series Theorem . . 316 10. Problems for Solution . . . . . . . . . . . 318 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CHAPTER X MARKOV PROCESSES AND SEMI-GROUPS . . . . . . . 1. The Pseudo-Poisson Type . . . . . . . . . . 2. A Variant: Linear Increments . . . . . . . . 3 . Jump Processes . . . . . . . . . . . . . 4. Diffusion Processes in R . . . . . . . . . . 5. The Forward Equation. Boundary Conditions . . . 1 321 322 324 326 332 337 344 345 349 353 356 6 . Diffusion in Higher Dimensions . . 7 . Subordinated Processes . . . . 8 . Markov Processes and Semi-Groups . . . . . . . . . . . . . . . . . . 9 . The "Exponential Formula" of Semi-Group Theory 10. Generators. The Backward Equation . . . . CHAPTER . . . . XI RENEWAL THEORY . 358 . 1. The Renewal Theorem . . . . . . . . . . 358 2 . Proof of the Renewal Theorem . . . . . . . . 364 *3 . Refinements . . . . . . . . . . . . . 366 . . . . . . . . 368 4 . Persistent Renewal Processes t 5 . The Number N of Renewal Epochs . . . . . . 372 6. Terminating (Transient) Processes . . . . . . . 374 7 . Diverse Applications . . . . . . . . . . . 377 8. Existence of Limits in Stochastic Processes . . . . . 379 *9 . Renewal Theory on the Whole Line . 10. Problems for Solution . . . . . CHAPTER . . . . . . 380 . . . . . . 385 XII RANDOM WALKS IN R1 . . . . . . . . . . . 389 1. Basic Concepts and Notations . . . . . . . . 390 2 . Duality . Types of Random Walks . . . . . . . 394 3 . Distribution of Ladder Heights . Wiener-Hopf Factorization . . . . . . . . . . . . . . . 398 3a. The Wiener-Hopf Integral Equation . . . . . . . 402 4 . Examples . . . . . . 5. Applications . . . . . 6. A Combinatorial Lemma . . 7. Distribution of Ladder Epochs 8. The Arc Sine Laws . . . . 9. Miscellaneous Complements . 10. Problems for Solution . . . CHAPTER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 408 412 413 417 423 425 XIII LAPLACE TRANSFORMS. TAUBERIAN THEOREMS. RESOLVENTS . 429 1. Definitions. The Continuity Theorem . . . . . . 429 2 . Elementary Properties . . . . . . . . . . . 434 3. Examples . . . . . . . . . . . . . . 436 4 . Completely Monotone Functions . Inversion Formulas . 439 5. Tauberian Theorems . . . . . . . . . . . 442 *6. Stable Distributions . . . . . . . . . . . 448 *7. Infinitely Divisible Distributions . . . . . . . . 449 *8. Higher Dimensions . . . . . . . . . . . 452 9. Laplace Transforms for Semi-Groups . . . . . . 454 10. The Hille-Yosida Theorem . . . . . . . . . 458 11. Problems for Solution . . . . . . . . . . . 463 CHAPTER XIV APPLICATIONS 470 473 475 5. Diffusion Processes . . . . . . . . . 479 6 . Birth-and-Death Processes and Random Walks . 483 7 The Kolmogorov Differential Equations . . . 488 8. Example: The Pure Birth Process . . . . . 9. Calculation of Ergodic Limits and of First-Passage Times 49 1 10. Problems for Solution . . . . . . . . . . . 495 . . . . 1. The Renewal Equation: Theory . . . . . . 2 . Renewal-Type Equations: Examples . . . . 3. Limit Theorems Involving Arc Sine Distributions . 4 . Busy Periods and Related Branching Processes . OF LAPLACE TRANSFORMS . . 466 . . 466 . . 468 . . . . . . . . . . . . . CHAPTER XV CHARACTERISTIC FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498 498 502 505 507 511 515 . . . . 2. Special Distributions . Mixtures . . . 2a . Some Unexpected Phenomena . . 3 . Uniqueness . Inversion Formulas . . 4. Regularity Properties . . . . . . 1. Definition. Basic Properties 5 . The Central Limit Theorem for Equal Components . 6 . The Lindeberg Conditions . . . . . . . . 7 . Characteristic Functions in Higher Dimensions . . *8 . Two Characterizations of the Normal Distribution 9 . Problems for Solution . . . . . . . . . CHAPTER . . 518 52 1 525 526 XVI* EXPANSIONS RELATED 1. Notations TO THE CENTRAL LIMIT THEOREM . . . . . . . . . . . . . 531 532 533 536 538 542 546 548 . . . . . . . . 2. Expansions for Densities . . . . 3 . Smoothing . . . . . . . . 4 . Expansions for Distributions . . . 5 . The Berry-Essten Theorems . . . . . . . . . . . . . . . . . . . . . . . 6 . Expansions in the Case of Varying Components . . . 7 . Large Deviations . . . . . . . . . . . . CHAPTER XVII INFINITELY D IVISIBLE DISTRIBUTIONS. 2a . Derivatives of Characteristic Functions . 3 . Examples and Special Properties . . . 4 . Special Properties . . . . . . . *6. Stable Densities 7 . Triangular Arrays . . . . . . . 554 1. Infinitely Divisible Distributions . . . . . . . . 554 2. Canonical Forms . The Main Limit Theorem . . . . 558 . . . . . . . . . . . . . . 5 . Stable Distributions and Their Domains of Attraction . . . . . . . . . . . . . . . . . . . . . . 565 566 . 570 . 574 . 581 . 583 . . . . . . . *9. Partial Attraction. "Universal Laws" *10. Infinite Convolutions . . . . . 11. Higher Dimensions . . . . . 12. Problems for Solution . . . . . CHAPTER *8 . The Class L . . . . . . . . . . . . . . . . . . . . . . . . . 588 590 592 593 595 . . . . . . XVIII APPLICATIONS OF FOURIER METHODS TO RANDOM WALKS . 598 598 601 604 609 612 614 . . . . . 616 . . . . . . . . *2. Finite Intervals. Wald's Approximation . 3 . The Wiener-Hopf Factorization . . . . 4 . Implications and Applications . . . . 5. Two Deeper Theorems . . . . . . 6 . Criteria for Persistency . . . . . . 7 . Problems for Solution . . . . . . . CHAPTER 1. The Basic Identity . . . . . . . . . . . . . . . . . . . . . . . . XIX HARMONIC ANALYSIS . . . . . . 1. The Parseval Relation . . . . . 2. Positive Definite Functions . . . 3 . Stationary Processes . . . . . 4 . Fourier Series . . . . . . . * 5 . The Poisson Summation Formula . 6. Positive Definite Sequences . . . 7. L 2 Theory . . . . . . . . 8 Stochastic Processes and Integrals . 9. Problems for Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619 619 620 623 626 629 633 635 641 647 ANSWERS TO PROBLEMS . . . . . . . . . . . . . . 651 SOME BOOKSON COGNATE SUBJECTS . . . . . . . . . . 655 INDEX . . . . . . . . . . . . . . . . . . . 657