Mathematics

The preliminary exam in stochastics is based on the material of the graduate probability sequence Math 523-524. The sequence covers standard topics of probability theory, starting with a brief introduction to measure theory foundations of probability and ending on the martingales theory. Martingale theory is fundamental to stochastic analysis, stochastic PDEs, mathematical finance, stochastic modeling and optimization and other applications. Therefore, there are many directions that a student can pursue after mastering material from this sequence. The current textbook is [4], other texts used in the past are in the References.

Foundations

  1. Basics from Measure Theory: Operations on sets; Collections of sets, σ-algebras and their Generators.).
  2. The Probability Space: Axioms of probability and basic formulas; The Borel sets on the real line; The Borel sets on \( \mathbb{R}^{n} \).

Random Variables

  1. Definition and Basic Properties: Functions of Random Variables; Operations on random variables; Approximation by simple random variables.
  2. Distributions: Distribution Functions; Decomposition of Distributions; Some Standard Discrete Distributions; Some Standard Absolutely Continuous Distributions;
  3. Random Vectors and Random Elements: Definitions and Characterization.

Expectation

  1. Construction and properties of expectation of a random variable
  2. Three theorems: Monotone Convergence Theorem, Dominated Convergence Theorem, and Fatou Lemma.
  3. Approximation by simple random variables argument
  4. A Change of Variables Formula
  5. Moments, Mean, Variance

Independence

  1. Independence of collections of events
  2. Independence of random variables and of their a-algebras
  3. Independence of functions of independent random variables
  4. Independence between collections of random variables
  5. Criteria for the independence of discrete and absolutely continuous random variables
  6. Kolmogorov’s zero-one law

Borel-Cantelli Lemmas

  1. Events occurring “infinitely often” or “eventually”
  2. Borcl-Cantclli Lemmas and their applications

Product Spaces; Fubini’s Theorem

  1. Finite and σ-finite measures
  2. Product measure and Fubini’s Theorem in connection to the independence (joint distributions, iterated expectations)
  3. Applications: Evaluation of expectations of functions of independent random variables; Expectation in terms of distribution function.

Inequalities

  1. Tail Probabilities Estimated by Moments: Markov’s and Chebyshev’s inequalities and the method to establish them.
  2. Moment Inequalities: The Holder inequality; The Minkowski inequality; Jensen’s inequality.
  3. Evaluation of the variance of a linear combination of random variables; covariance and correlation bounds.

Convergence

  1. Four types of convergence of random variables: Almost Sure; in Probability; in the pth mean (in \( L^{p} \)); in Distribution.
  2. Relations between the convergence almost surely and in probability. Measurability and uniqueness of limits. A subsequence principle.
  3. Relations between the Four types of convergence of random variables.
  4. Cauchy Convergence; Uniform Integrability; Convergence of Moments
  5. Convergence of Sums of Sequences: Slutsky’s theorem

The Law of Large Numbers

  1. Preliminaries: Convergence Equivalence and Distributional Equivalence
  2. The Weak Law of Large Numbers
  3. The Three-Series Theorem
  4. The Strong Law of Large Numbers
  5. Applications

Characteristic functions

  1. Definition, basic properties, characteristic functions of some standard distributions
  2. Uniqueness and Inversion formulas
  3. Characteristic function of the sum of independent random variables
  4. Taylor polynomial approximation of a characteristic function
  5. Moments of a random variable by differentiating its characteristic function
  6. Characteristic Function of a Random Vector (multivariate characteristic function)
  7. Independence of random variables via multivariate characteristic function
  8. Multivariate normal distribution

Convergence in distribution revisited

  1. Convergence in distribution and Tightness
  2. Lévy’s Convergence Theorem
  3. The Continuous Mapping Theorem

The Central Limit Theorem (CLT)

  1. Method of characteristic function
  2. CLT for i.i.d. random variables
  3. Applications

Martingales

  1. Conditional expectation: Definitions, properties, composition of conditional expectations.
  2. Martingales: Definitions and examples; Orthogonality of martingale differences
  3. Doob’s decomposition of submartingales
  4. Stopping times: Definition and properties; randomly stopped martingales
  5. The optional sampling theorem
  6. Maximal inequalities and Martingale convergence theorem
  7. Martingale convergence for \( E(Z|\mathscr{F}_n) \) and the uniform integrability
  8. Stopped random walk and the Wald equations

References

[1] Billingsley, P. Probability and Measure, 3rd ed. Wiley, (1995).
[2] Durrett, R. Probability: Theory and Examples, 4th ed. Cambridge University Press, (2010).
[3] Fristedt, B. and Grey, L. A Modem Approach to Probability Theory, Birkhauser, (1997).
[4] Gut, A. Probability: A Graduate Course. 2nd ed. Springer, (2013).
[5] Jacod, J. and Protter, P. Probability Essentials, 2nd ed. Springer, (2004).