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An Introduction to probability theory and its applications

Author: Feller, William Publisher: Wiley, 1971.Edition: 2nd ed.Language: EnglishDescription: 24 cm.ISBN: 0-471-25709-5Type of document: BookNote: Doriot and Tanoto: for 2016-2017 coursesBibliography/Index: Includes bibliographical references List(s) this item appears in: Textbooks for Discrete Stochastic Processes / Stephen Chick / PhD 2016-2017 | Textbooks for Probability and Statistics A / Spyros Zoumpoulis / PhD 2016-2017
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Item type Current location Collection Call number Status Date due
Book Doriot Library
Main Collection
Print QA273 .F45 1971 Vol.2
(Browse shelf)
001352862
Available
Book Doriot Library
Main Collection
Print QA273 .F45 1971 Vol.2
(Browse shelf)
000127245
Available
Book Tanoto Library
Textbook Collection (PhD)
Print QA273 .F45 1971 Vol.2
(Browse shelf)
900041496
Consultation only
Book Tanoto Library
Textbook Collection (PhD)
Print QA273 .F45 1971 Vol.2
(Browse shelf)
900203397
Available

Doriot and Tanoto: for 2016-2017 courses

Includes bibliographical references

Digitized

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

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