Probability and Stochastic Processes

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