Probability — Advanced

Stochastic Processes and Applications (Graduate Level)

Overview of the Advanced level

The Advanced level covers the theory and applications of stochastic processes—a mathematical framework for handling random phenomena that evolve over time.

Learning goals

  • Understand the basic concepts of stochastic processes
  • Master the theory of Markov chains
  • Understand the construction and properties of Brownian motion
  • Understand the representation of stochastic processes via the Karhunen-Loève expansion
  • Learn the basics of the Itô integral and stochastic differential equations
  • Be able to prove the central limit theorem
  • Understand the foundations of information theory

Contents

  1. Chapter 1 Stochastic Processes

    Definition of a stochastic process, finite-dimensional distributions, sample paths

  2. Chapter 2 Markov Processes

    Markov property, transition probability matrix, stationary distribution, ergodicity

  3. Chapter 3 Brownian Motion

    Wiener process, continuity and non-differentiability of paths, reflection principle

  4. Chapter 4 Karhunen-Loève Expansion

    Optimal orthogonal expansion of stochastic processes, eigenvalue problem of the covariance operator, relation to PCA

  5. Chapter 5 Stochastic Differential Equations

    Itô integral, Itô's formula, solutions of SDEs

  6. Chapter 6 Central Limit Theorem

    Characteristic functions, proof of the central limit theorem

  7. Chapter 7 Information Theory

    Entropy, mutual information, the source coding theorem

  8. Chapter 8 Modern Topics

    Martingales, the large deviation principle, introduction to random matrices

Martingale Topics

  1. Stopping Time

    Definition of a stopping time, relation to filtrations, and the bridge to the optional stopping theorem.

  2. Optional Stopping Theorem

    Conditions for the optional stopping theorem, a proof sketch, and its application to the gambler's ruin problem.

  3. Wald's Identity

    The expected value of a random sum, $E[S_N]=\mu E[N]$, and its application to sequential analysis.

  4. Azuma's Inequality

    Concentration of bounded-increment martingales, generalizing the method of bounded differences.

Distribution Reference

  1. Stable Distribution

    Definition, characteristic function, mean, variance, special cases, and applications of the stable distribution.

  2. q-Gaussian Distribution

    Definition, PDF, mean, variance, and its connection to Tsallis statistics.

  3. Mittag-Leffler Distribution

    Definition, PDF, mean, variance, and the role of the Mittag-Leffler function.

  4. Tukey Lambda Distribution

    Definition, quantile function, mean, variance, and applications.

  5. Wigner Semicircle Distribution

    Definition, PDF, mean, variance, and its role in random matrix theory.

  6. Nakagami Distribution

    Definition, PDF, mean, variance, and applications to fading channels.

  7. Rice Distribution

    The probability density function (with the modified Bessel function $I_0$), parameters, and applications.

  8. Burr Distribution

    Definition, PDF, CDF, mean, variance, and applications.

  9. Dagum Distribution

    Definition, PDF, CDF, mean, variance, income distribution, and relation to the Burr distribution.

  10. Lomax Distribution

    Definition, PDF, CDF, mean, variance, and applications (Pareto Type II).

  11. Gompertz Distribution

    A survival-time distribution with an exponentially increasing hazard function.

  12. Kumaraswamy Distribution

    A flexible continuous distribution on the interval $[0,1]$.

  13. Irwin-Hall Distribution

    Definition, PDF, mean, variance, and connection to the central limit theorem.

  14. Bates Distribution

    Definition, PDF, mean, variance, and relation to the Irwin-Hall distribution.

  15. Delaporte Distribution

    Definition, PMF, mean, variance, and applications (a Poisson-negative-binomial mixture).

  16. Conway-Maxwell-Poisson Distribution

    A generalized Poisson distribution that handles over- and under-dispersion.

  17. Beta Prime Distribution

    Definition, PDF, mean, variance, and relation to the beta distribution.

  18. Exponentially Modified Gaussian Distribution

    The convolution of a Gaussian and an exponential distribution.

  19. Hyperbolic Secant Distribution

    Definition, PDF, mean, variance, and characteristic properties.

  20. Logarithmic Distribution

    Definition, PMF, mean, variance, and applications (the logarithmic series distribution).

  21. Lévy Distribution

    Definition, PDF, mean and variance, its property as a stable distribution, and applications.

  22. Skellam Distribution

    Definition, PMF, mean, variance, and applications (the difference of two independent Poisson variables).

  23. Slash Distribution

    Definition, PDF, mean, variance, and applications (a standard normal divided by a uniform variable).

  24. Variance-Gamma Distribution

    Definition, PDF, mean, variance, skewness, and applications.

  25. Yule-Simon Distribution

    Definition, PMF, mean, variance, and applications.

  26. Zeta Distribution

    Definition, PMF, mean, variance, and relation to the Riemann zeta function (Zipf's law).

  27. Raised Cosine Distribution

    Definition, PDF, CDF, mean, and variance.

  28. Negative Hypergeometric Distribution

    Definition, PMF, mean, and variance.

  29. Rademacher Distribution

    Definition, PMF, mean, variance, and applications of the $\pm 1$ two-valued distribution.

Reading

  • Continuous Everywhere, Differentiable Nowhere [Reading]

    Born from the dance of pollen grains, Brownian motion is a monster that is continuous yet nowhere differentiable. The story of the heart of probability theory, tackled by Einstein, Wiener, and Itô.

  • The Mathematics of a Fair Game [Reading]

    In a fair game, no betting strategy can increase your winnings. From the trap of the double-or-nothing strategy to the optional stopping theorem—the story of martingales, which made "there is no clever trick" rigorous.

Prerequisites