Math 563 - Topics
The nominal text for the course is
A Modern Approach To Probability Theory by Fristedt and Gray.
The following is a subset of the table of contents of Fristedt and Gray.
I hope to cover at least some of each of these chapters, but
some of the sections will have to be skipped.
Some of the sections are a review of measure theory and integration
and will be covered rather quickly.
1 Probability spaces
1.1 Introductory examples
1.2 Ingredients of probability spaces
1.3 sigma-fields
1.4 Borel sigma-fields
2 Random variables
2.1 Defs and basic results
2.2 R^d valued random variables
2.3 R^infinity valued random variables
2.4 Further examples
3 Distribution functions
3.1 Basic theory
3.2 Examples of distributions
3.3 Some descriptive terminology
3.4 Distributions with densities
3.5 Further examples
3.6 Distribution functions for the extended real line
4 Expectation theory
4.1 Defs
4.2 Linearity and positivity
4.3 Monotone convergence
4.4 Expectation of compositions
4.5 The Riemann-Stieltjes integral and expectations
5 Expectations: applications
5.1 Variance and the Law of Large Numbers
5.2 Mean vectors and covariance matrices
5.3 Moments and the Jensen inequality
5.4 Probability generating functions
5.5 Characterization of probability generating functions
6 Calculating probabilities and measures
6.1 Operations on events
6.2 Borel-Cantelli lemma and Kochen-Stone lemma
6.3 Inclusion-exclusion
6.4 Finite and sigma-finite measures
7 Measure theory : existence and uniqueness
7.1 Sierpinski class thm and uniqueness
7.2 Finitely additive functions defined on fields
7.3 Existence, extension and completion of measures
7.4 Examples
8 Integration theory
8.1 Lebesgue integration
8.2 Convergence theorems
8.3 Probability measures and infinite measures compared
8.4 Lebesgue integrals and Riemann-Stieltjes integrals
8.5 Absolute continuity and densities
8.6 Integration with respect to counting measure
9 Stochastic independence
9.1 Def and basic properties
9.2 Product measure: finitely many factors
9.3 Fubini's theorem
9.4 Expectations and independence
9.5 Densities and independence
9.6 Product probability measures - infinitely many factors
9.7 Borel-Cantelli lemma
10 Sums of independent random variables
10.1 Convolutions of distributions
10.2 Multinomial distributions
10.3 Probability generating functions
12 Theorems of a.s. convergence
12.1 Convergence in probability
12.2 Laws of large numbers
12.3 Applications
12.4 0-1 laws
12.5 Random infinite series
12.6 Etemadi lemma
12.7 Kolmogorov three series theorem
13 Characteristic functions
13.1 Def and basic examples
13.2 The Parsaval relation and uniqueness
13.3 Characteristic functions of convolutions
13.4 Symmetrization
13.5 Moment generating functions
13.6 Moment theorems
13.7 Inversion theorem
13.8 Characteristic functions in R^d
13.9 Normal distributions on R^d
14 Convergence in distribution on R
14.1 Defs and examples
14.2 Limit distributions for extreme values
14.3 Relationships to other types of convergence
14.4 Convergence conditions for sequences of distributions
14.5 Sequences of distributions on extended real line
14.6 Relative sequential compactness
14.7 The continuity theorem
14.8 Scaling and centering of sequences of distributions
14.9 Characterization of moment generating functions
14.10 Characterization of characteristic functions
15 Distributional limit theorems for partial sums
15.1 Infinite series of independent random variables
15.2 The law of large numbers revisted
15.3 The classical central limit theorem
15.4 The general setting for i.i.d. sequences
15.5 Large deviations
20 Spaces of random variables
20.1 Hilbert spaces
20.2 Hilbert space L^2
20.3 Metric space L^1
20.4 Best linear estimator
21 Conditional probabilities
21.1 The construction of conditional probabilities
21.2 Conditional distributions
21.3 Conditional densities
21.4 Existence and uniqueness of conditional distributions
21.5 Conditional independence
22 Construction of random sequences
22.1 The basic result
22.2 Construction of exchangeable sequences
22.3 Construction of Markov sequences
22.4 Polya urns
23 Conditional expectation
23.1 Def of conditional expectation
23.2 Conditional versions of unconditional theorems
23.3 Formulas for conditional expectations
23.4 Conditional variance