Probability and Stochastic Processes

List of Figures xvii List of Tables xxi Preface i Acknowledgments iii Introduction 1 PART I PROBABILITY 1 Elements of Probability Measure 3 1.1 Probability Spaces 4 1.2 Conditional Probability 16 1.3 Independence 23 1.4 Monotone Convergence properties of probability 25 1.5 Lebesgue measure on the unit interval (0,1] 31 Problems 34 2 Random Variables 39 2.1 Discrete and Continuous Random Variables 42 2.2 Examples of commonly encountered Random Variables 46 2.3 Existence of random variables with prescribed distribution. Skorohod representation of a random variable 59 2.4 Independence 62 2.5 Functions of random variables. Calculating distributions 66 Problems 76 3 Applied chapter: Generating Random Variables 81 3.1 Generating one dimensional random variables by inverting the CDF 82 3.2 Generating one dimensional normal random variables 85 3.3 Generating random variables. Rejection sampling method 88 3.4 Generating random variables. Importance sampling 104 Problems 113 4 Integration Theory 117 4.1 Integral of measurable functions 118 4.2 Expectations 124 4.3 Moments of a random variable. Variance and the correlation coefficient. 137 4.4 Functions of random variables. The Transport Formula. 139 4.5 Applications. Exercises in probability reasoning. 142 4.6 A Basic Central Limit Theorem: The DeMoivre-Laplace Theorem: 144 Problems 146 5 Conditional Distribution and Conditional Expectation 149 5.1 Product Spaces 150 5.2 Conditional distribution and expectation. Calculation in simple cases 154 5.3 Conditional expectation. General definition 157 5.4 Random Vectors. Moments and distributions 160 Problems 169 6 Moment Generating Function. Characteristic Function. 173 6.1 Sums of Random Variables. Convolutions 173 6.2 Generating Functions and Applications 174 6.3 Moment generating function 180 6.4 Characteristic function 184 6.5 Inversion and Continuity Theorems 191 6.6 Stable Distributions. Lévy Distribution 196 Problems 200 7 Limit Theorems 205 7.1 Types of Convergence 205 7.2 Relationships between types of convergence 213 7.3 Continuous mapping theorem. Joint convergence. Slutsky's theorem 222 7.4 The two big limit theorem: LLN and CLT 224 7.5 Extensions of CLT 237 7.6 Exchanging the order of limits and expectations 243 Problems 244 8 Statistical Inference 251 8.1 The classical problems in statistics 251 8.2 Parameter Estimation problem 252 8.3 Maximum Likelihood Estimation Method 257 8.4 The Method of Moments 268 8.5 Testing, The likelihood ratio test 269 8.6 Confidence Sets 276 Problems 278 PART II STOCHASTIC PROCESSES 9 Introduction to Stochastic Processes 285 9.1 General characteristics of Stochastic processes 286 9.2 A Simple process? The Bernoulli process 293 Problems 296 10 The Poisson process 299 10.1 Definitions. 299 10.2 Interarrival and waiting time for a Poisson process 302 10.3 General Poisson Processes 309 10.4 Simulation techniques. Constructing Poisson Processes 315 Problems 318 11 Renewal Processes 323 11.1 Limit Theorems for the renewal process 326 11.2 Discrete Renewal Theory. 335 11.3 The Key Renewal Theorem 340 11.4 Applications of the Renewal Theorems 342 11.5 Special cases of renewal processes 344 11.6 The renewal Equation 350 11.7 Age dependent Branching processes 354 Problems 357 12 Markov Chains 361 12.1 Basic concepts for Markov Chains 361 12.2 Simple Random Walk on integers in d-dimensions 373 12.3 Limit Theorems 376 12.4 States in a MC. Stationary Distribution 377 12.5 Other issues: Graphs, first step analysis 384 12.6 A general treatment of the Markov Chains 385 Problems 395 13 Semi-Markov and Continuous time Markov Processes 401 13.1 Characterization Theorems for the general semi Markov process 403 13.2 Continuous time Markov Processes 407 13.3 The Kolmogorov Differential Equations 410 13.4 Calculating transition probabilities for a Markov process. General Approach 415 13.5 Limiting Probabilities for the Continuous time Markov Chain 416 13.6 Reversible Markov process 419 Problems 422 14 Martingales 427 14.1 Definition and examples 428 14.2 Martingales and Markov chains 430 14.3 Previsible process. The Martingale transform 432 14.4 Stopping time. Stopped process 434 14.5 Classical examples of Martingale reasoning 439 14.6 Convergence theorems. L1 convergence. Bounded martingales in L2 446 Problems 448 15 Brownian Motion 455 15.1 History 455 15.2 Definition 457 15.3 Properties of Brownian motion 461 15.4 Simulating Brownian motions 470 Problems 471 16 Stochastic Differential Equations 475 16.1 The construction of the stochastic integral 477 16.2 Properties of the stochastic integral 484 16.3 Ito lemma 485 16.4 Stochastic Differential equations. SDE's 489 16.5 Examples of SDE's 492 16.6 Linear systems of SDE's 503 16.7 A simple relationship between SDE's and PDE's 505 16.8 Monte Carlo Simulations of SDE's 507 Problems 512 A Appendix: Linear Algebra and solving difference equations and systems of differential equations 517 A.1 Solving difference equations with constant coefficients 518 A.2 Generalized matrix inverse and pseudodeterminant 519 A.3 Connection between systems of differential equations and matrices520 A.4 Linear Algebra results 523 A.5 Finding fundamental solution of the homogeneous system 526 A.6 The nonhomogeneous system 528 A.7 Solving systems when P is nonconstant 530 Index 533