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Probability theory with applications

Author: Rao, M. M. ; Swift, R. J. Series: Mathematics and its applications ; 582 Publisher: Springer, 2006.Edition: 2nd ed.Language: EnglishDescription: 527 p. : Graphs ; 24 cm.ISBN: 0387277307Type of document: BookBibliography/Index: Includes bibliographical references and index
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Item type Current location Collection Call number Status Date due Barcode Item holds
Book Europe Campus
Main Collection
Print QA273 .R36 2006
(Browse shelf)
001175508
Available 001175508
Total holds: 0

Includes bibliographical references and index

Digitized

Probability Theory with Applications Contents Preface to Second Edition ................................................................... ix Preface to First Edition ...................................................................... xv List of Symbols...................................................................................... xvii Part I - Foundations.............................................................................. 1 1 Background Material and Preliminaries............................................... 3 1.1 What is Probability?................................................................................................................................ 3 1.2 Random Variables and Measurability Results ................................ 7 1.3 Expectations and the Lebesgue Theory......................................... 12 1.4 Image Measure and the Fundamental Theorem of Probability ...... 20 Exercises .................................................................................. 28 2 Independence and Strong Convergence ............................................ 2.1 Independence ............................................................................. 2.2 Convergence Concepts, Series and Inequalities............................ 2.3 Laws of Large Numbers ............................................................... 2.4 Applications to Empiric Distributions, Densities, Queueing, and Random Walk...................................................................... Exercises .................................................................................. 33 33 46 58 68 87 3 Conditioning and Some Dependence Classes .................................... 103 3.1 Conditional Expectations............................................................ 103 3.2 Conditional Probabilities ........................................................... 120 3.3 Markov Dependence ................................................................... 140 3.4 Existence of Various Random Families ....................................... 158 3.5 Martingale Sequences ................................................................ 174 Exercises ..................................................................................203 Part II - Analytical Theory..................................................................... 221 4 Probability Distributions and Characteristic Functions ................... 223 4.1 Distribution Functions and the Selection Principle ..................... 223 4.2 Characteristic Functions, Inversion, and Lévy's Continuity Theorem ................................................................................... 234 4.3 Cramér's Theorem on Fourier Transforms of Signed Measures ... 251 4.4 Bochner's Theorem on Positive Definite Functions ...................... 256 viii Contents 4.5 Some Multidimensional Extensions ............................................ 265 4.6 Equivalence of Convergences for Sums of Independent Random Variables ..................................................................... 274 Exercises .................................................................................. 276 5 Weak Limit Laws................................................................................. 291 5.1 Classical Central Limit Theorems................................................. 291 5.2 Infinite Divisibility and the Lévy-Khintchine Formula .................. 304 5.3 General Limit Laws, Including Stability ....................................... 318 5.4 Invariance Principle..................................................................... 341 5.5 Kolmogorov's Law of the Iterated Logarithm................................. 364 5.6 Application to a Stochastic Difference Equation .......................... 375 Exercises .................................................................................. 386 Part III - Applications............................................................................ 409 6 Stopping Times, Martingales, and Convergences................................ 411 6.1 Stopping Times and Their Calculus..............................................411 6.2 Wald's Equation and an Application ............................................415 6.3 Stopped Martingales.................................................................... 420 Exercises .................................................................................. 427 7 Limit Laws for Some Dependent Sequences......................................... 429 7.1 Central Limit Theorems............................................................... 429 7.2 Limit Laws for a Random Number of Random Variables............... 436 7.3 Ergodic Sequences.......................................................................449 Exercises .................................................................................. 455 8 A Glimpse of Stochastic Processes...................................................... 459 8.1 Brownian Motion: Definition and Construction ........................... 459 8.2 Some Properties of Brownian Motion ...........................................463 8.3 Law of the Iterated Logarithm for Brownian Motion ..................... 467 8.4 Gaussian and General Additive Processes.................................... 470 8.5 Second-Order Processes ............................................................. 493 Exercises .................................................................................. 498 References............................................................................................. 509 Author Index......................................................................................... 519 Subject Index ........................................................................................ 523

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