A Standard Introduction to Probability
Math 581
Fall 2006

Week 2 : Binomial, Multinomial and Hypergoemetric Distributions, Bernoulli Trials, Random Walk on Lattices and Asymptotics, First Limit Theorems (Theorem of Poisson 1887), Probability Spaces on Euclidean Spaces, Probability Distribution in R^1, Probability Density in R^1, Probability Spaces on Euclidean Spaces R^d, Probability Distribution in R^d, Properties and Identification with Measures, Lebesgue's Decomposition, Radon-Nikodym Theorem and Derivatives, Probability Densities in R^d, Examples (Gamma, Exponential, Weibull, Maxwell, Planck, Normal, Log-Normal, Uniform, t-, Chi^2-, Beta, Gauss,...), Multidimensional Distributions, Key Properties, Equivalence of Measures - Distributions - Densities

Week 3 : Conditional Probabilities, Formulas of Total Probability and Bayes, Multiplicative Theorem, Aposteriori Quality Control, Pairwise Independence, Complete Independence, Bernoulli-Scheme, Independence of Sigma-Algebras, Dynkin Systems, Naturally Generated Sigma-Algebras, Construction of Independent Sigma-Algebras Using Dynkin Systems, Terminal Events, Terminal Sigma-Algebra, 0-1 Law of Kolmogorov

Week 4 : 0-1 Law of Borel, 0-1 Law of Borel-Cantelli, Product Probability Spaces, Product Sigma Algebras, Product Measures, Cylindrical Sets, Extension Theorem of Kolmogorov (Family of Finite-Dimensional Distributions), Random Variables as Measurable Mappings, Characteristics (Induced Distributions, Densities, Probability Spaces), Transformation of Random Variables, Distributions Described by Lebesgue-Stieltjes Integrals, Relation Riemann-Stieltjes and Lebesgue-Stieltjes Integrals, Functions of Bounded Total Variation

Week 5 : Random Vectors, Marginal Distributions and Marginal Densities, Independence of Random Variables and Random Vectors, Transformation of Random Vectors, Products of Random Variables, Sums of Random Variables, Convolution, Properties of Convolutions, Convolution of Gaussian Variables, Convolution of Densities, Convolution of Independent Distributions, Theory of Expectation, Basic Properties, Linearity and Monotonicity of Expectation, Examples, Normalization Constant of Gaussian Random Variables, Jensen's Inequality, Review on Convex Functions

Week 6 : Basic Probabilistic Inequalities (Lyapunov, Cauchy-Bunyakovski-Schwarz, Hoelder, Minkowski), Continuous Embedding of L^p-Spaces of Random Variables, P-integrability and Expectation, Moments, Expansion of Central Moments, Classical Moment Problem, Criterion of Carleman, Variance, Standard Deviation, Basic Variance Properties, Minimizing Property and Best L^2-Approximation, Standardized, Normalized and Centered Random Variables, Moments of Gaussian Distribution

Week 7 : Effect of Independence on Variance Computations, Weak Independence, Estimation of Probabilities From Above, Chebyshev-Markov Inequality, Estimation of Probabilities From Below, Diverse Special Cases, Application to Constructive Proof of Weierstrass Approximation Theorem, Loeves Basic Inequality, Independence, Correlation Measures, Covariance Matrices, Basic Properties, Classification by Correlation Coefficients

Week 8 : Further Properties of Covariance Matrices, Examples, L^p-Theory of Random Variables, L^p and Hilbert Spaces of Random Variables, Completeness, Monotone Convergence Theorem, Lemma of Fatou, Lebesgue's Theorem on Dominated L^p-Convergence, Uniform p-Integrability, Hilbert-spaces and Projections, Best Linear Estimators of Random Variables, Orthogonality, Orthonomal Basis, Separability, Parsevals Identity, and Bessels Inequality, Conditional Expectation as Projection, Continuity of Scalar Product, Properties of Projections, Orthogonal Decompositions, Gaussian Spaces and Hermitian Polynomials, Poisson Space

Week 9 : Characteristic Functions as Fourier Transforms, Basic Properties, Residue Theorem and Parseval's Identity for Computations, Uniform Continuity, Taylor Expansions and Semiinvariants, Cumulant-Functions, Differentiability and Moments, Uniqueness Theorem, Independence, Inverse Transformations and Formula, Consequences, Lattice Distributions, Convolution Theorem, Continuity Theorem, Bochner-Khinchin Characterization on Positive-Definiteness, Theorem of Riemann-Lebesgue (Behavior at Infinity), Polya's Convex-Characterization, Marcinkiewicz's Theorem on Exponential-Polynomial Growth, Discreteness,

Week 10 : Gaussian Systems, Linear Gaussian Transformations, Theorem of Fisher on Invariance of Orthogonal Transformations, Theorem of Sinai on Symmetric Estimation, Infinitely Devisible Distributions, Examples, Levy-Khinchin Theorem, The Example of Cauchy-Distribution, Kolmogorov-Representation, Properties of I.D. Distributions, Stable Distributions, Levy-Khinchin-Representation, Finiteness of Moments, Nondifferentiability For Stable Class

Week 11 : Conditional Expectation Value Given Event B, Conditional Probability Distribution, Discrete Conditional Expectation, The Example of Exponential Distributions, Memoryless Property, The Example of Serving Machines

Week 12 : Conditional Expectations via Radon-Nikodym Theorem, L^2-Projections and Minimizing Property, Orthogonal Decompositions, Linearity and Tower Property, Jensens Inequality, Conditional Distribution Functions, Conditional Densities, Conditional Probability Given F_0, Gaussian Conditional Expectations, Linearity w.r.t. Conditions, Explicit Series Representation, A Formal Algorithm to Compute Conditional Expectations

Week 13 : Discrete Martingale Theory, Submartingales, Supermartingales, Filtration, Monotonicity of Moments, Generation by Independent Increments, Examples, Doob-Levy Martingale, Variance Martingales, Product Martingales, De Moivre-Martingale (Gamblers Ruin), Harmonic Functions and Martingales, Exponential Martingales

Week 14 : Large Deviations and Hoeffdings Inequality, Bernstein Inequality, Markov-Times and Stopping Times, Stopped Sigma-Algebras, Stopped Filtrations Optional Sampling Theorems, Weak and Strong Optional Sampling, Counterexample, Converse Theorems to Characterize Martingales, SubMartingales and Convex Transformations, Supermartingales and Concave Transformations, Doobs Maximal Inequalities (Doobs First Martingale Inequality, Doobs L^p-Martingale Inequality)

Week 15 : Convergence Concepts of Sequences of Random Variables (Almost sure, L^p, fast L^p, in probability, weak and in distribution), Limit Theorems (WLLN, SLLN, CLTs, De Moivre-Laplace, Lindenberg-Levy, Lindenberg-Feller, Lyapunov Condition), Berry-Esseen-Inequality, LiL, Fundamental Theorem of Mathematical Statistics (Glivenko-Cantelli)

Selected Further Topics Depending on Interest of Audience: Doobs Upcrossing Inequality, Doobs Martingale Convergence Theorem, Martingale Convergence Theorems, Application to Kolmogorovs SLLN, Supermartingale and Submartingale Convergence Theorems, Closable Sequences, Local Sub-, Super- and Martingales, Martingale Transformations, Predictable Sequences, Discrete Integral Transformations, Martingale Differences and Bernoulli Game, Generalized Kolmogorov Inequality, Doob-Meyer Decomposition and Compensators, Mean-Square Integrable Martingales, Quadratic Variation, Mutual Characteristics of X and Y, Quadratic Covariation, Discrete Ito Formula, Discrete Burkholder Inequality, Markov Chains & Markov Processes, Entropy and Information, Coding Theorems, Selected Advanced Topics (Stochastic Integration, Semimartingales, Ito's Lemma, Brownian Motion, Poisson Process, SDEs, SPDEs, Large Deviations, Monte Carlo Methods, Simulation of Random Variables and Stochastic Processes)

Read this Without Tears: WHAT IS EXPECTED OF YOU (From "Teaching at the University Level" by Stephen Zucker, Notices Amer. Math. Soc. 43 (1996), p. 863))

40 Further Introductory Readings To Probability and Stochastic Processes For The Ideal Student (optional, for those they want to have profound readings I recommend to work through this list, and make your personal preferences, but do not expect to understand them all immediately, it took me more than 16 years, and we are still working on it!):