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. Levy 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