.

0 (hodnocen0 x )

(0.5) Půjčeno:1x 

BK

3rd ed.

Oxford : Oxford University, 2001

xii, 596 s.

ISBN 0-19-857222-0 (brož.)

Obsahuje předmluvu, dodatky, rejstřík

Bibliografie na s. 580-582

Pravděpodobnost - učebnice vysokošk.

Procesy náhodné - učebnice vysokošk.

000087033

Contents // 1 Events and their probabilities // 1.1 Introduction 1 // 1.2 Events as sets 1 // 1.3 Probability 4 // 1.4 Conditional probability 8 // 1.5 Independence 13 // 1.6 Completeness and product spaces 14 // 1.7 Worked examples 16 // 1.8 Problems 21 // 2 Random variables and their distributions // 2.1 Random variables 26 // 2.2 The law of averages 30 // 2.3 Discrete and continuous variables 33 // 2.4 Worked examples 35 // 2.5 Random vectors 38 // 2.6 Monte Carlo simulation 41 // 2.7 Problems 43 // 3 Discrete random variables // 3.1 Probability mass functions 46 // 3.2 Independence 48 // 3.3 Expectation 50 // 3.4 Indicators and matching 56 // 3.5 Examples of discrete variables 60 // 3.6 Dependence 62 // 3.7 Conditional distributions and conditional expectation 67 // 3.8 Sums of random variables 70 // 3.9 Simple random walk 71 // 3.10 Random walk: counting sample paths 75 // 3.11 Problems 83 // x // Contents // 4 Continuous random variables // 4.1 Probability density functions 89 // 4.2 Independence 91 // 4.3 Expectation 93 // 4.4 Examples of continuous variables 95 // 4.5 Dependence 98 // 4.6 Conditional distributions and conditional expectation 104 // 4.7 Eunctions of random variables 107 // 4.8 Sums of random variables 113 // 4.9 Multivariate normal distribution 115 // 4.10 Distributions arising from the normal distribution 119 // 4.11 Sampling from a distribution 122 // 4.12 Coupling and Poisson approximation 127 // 4.13 Geometrical probability 133 // 4.14 Problems 140

// 5 Generating functions and their applications // 5.1 Generating functions 148 // 5.2 Some applications 156 // 5.3 Random walk 162 // 5.4 Branching processes 171 // 5.5 Age-dependent branching processes 175 // 5.6 Expectation revisited 178 // 5.7 Characteristic functions 181 // 5.8 Examples of characteristic functions 186 // 5.9 Inversion and continuity theorems 189 // 5.10 Two limit theorems 193 // 5.11 Large deviations 201 // 5.12 Problems 206 // 6 Markov chains // 6.1 Markov processes 213 // 6.2 Classification of states 220 // 6.3 Classification of chains 223 // 6.4 Stationary distributions and the limit theorem 227 // 6.5 Reversibility 237 // 6.6 Chains with finitely many states 240 // 6.7 Branching processes revisited 243 // 6.8 Birth processes and the Poisson process 246 // 6.9 Continuous-time Markov chains 256 // 6.10 Uniform semigroups 266 // 6.11 Birth-death processes and imbedding 268 // 6.12 Special processes 274 // 6.13 Spatial Poisson processes 281 6 14 Markov i’hain Monte ???» 9Q1 // Contents // 7 Convergence of random variables // 7.1 Introduction 305 // 7.2 Modes of convergence 308 // 7.3 Some ancillary results 318 // 7.4 Laws of large numbers 325 // 7.5 The strong law 329 // 7.6 The law of the iterated logarithm 332 // 7.7 Martingales 333 // 7.8 Martingale convergence theorem 338 // 7.9 Prediction and conditional expectation 343 // 7.10 Uniform integrability 350 // 7.11 Problems 354 // 8 Random processes // 8.1 Introduction 360 // 8.2 Stationary processes 361

// 8.3 Renewal processes 365 // 8.4 Queues 367 // 8.5 The Wiener process 370 // 8.6 Existence of processes 371 // 8.7 Problems 373 // 9 Stationary processes // 9.1 Introduction 375 // 9.2 Linear prediction 377 // 9.3 Autocovariances and spectra 380 // 9.4 Stochastic integration and the spectral representation 387 // 9.5 The ergodic theorem 393 // 9.6 Gaussian processes 405 // 9.7 Problems 409 // 10 Renewals // 10.1 The renewal equation 412 // 10.2 Limit theorems 417 // 10.3 Excess life 421 // 10.4 Applications 423 // 10.5 Renewal-reward processes 431 // 10.6 Problems 437 // 11 Queues // 11.1 Single-server queues 440 // 11.2 M/M/1 442 // 11.3 M/C/1 445 // 11.4 G/M/l 451 // 11.5 G/G/l 455 // xii // Contents // 11.6 Heavy traffic 462 // 11.7 Networks of queues 462 // 11.8 Problems 468 // 12 Martingales // 12.1 Introduction 471 // 12.2 Martingale differences and Hoeffding’s inequality 476 // 12.3 Crossings and convergence 481 // 12.4 Stopping times 487 // 12.5 Optional stopping 491 // 12.6 The maximal inequality 496 // 12.7 Backward martingales and continuous-time martingales 499 // 12.8 Some examples 503 // 12.9 Problems 508 // 13 Diffusion processes // 13.1 Introduction 513 // 13.2 Brownian motion 514 // 13.3 Diffusion processes 516 // 13.4 First passage times 525 // 13.5 Barriers 530 // 13.6 Excursions and the Brownian bridge 534 // 13.7 Stochastic calculus 537 // 13.8 The Itô integral 539 // 13.9 Itô’s formula 544 // 13.10 Option pricing 547 // 13.11 Passage probabilities

and potentials 554 // 13.12 Problems 561 // Appendix I. Foundations and notation 564 // Appendix II. Further reading 569 // Appendix III. History and varieties of probability 571 // Appendix IV. John Arbuthnot’s Preface to Of the laws of chance (1692) // Appendix V. Table of distributions 576 // Appendix VI. Chronology 578 // Bibliography 580 // Notation 583 // Index 585