Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Martingale Theory: Basics and Applications

In course

MATH-467: Probabilistic methods in combinatorics

The 'probabilistic method' is a fundamental tool in combinatorics. The basic idea is as follows: to prove that an object (for example, graph) with certain properties exists, it suffices to prove that if the object is chosen at random, then it has the desir ...

Description

This lecture covers the basics of Martingale theory, including conditional expectations, law of total expectations, and Martingale properties. It also explores applications of Martingales in random variables and provides corollaries for more general situations.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Official source

Related concepts (21)

Law of total expectation

The proposition in probability theory known as the law of total expectation, the law of iterated expectations (LIE), Adam's law, the tower rule, and the smoothing theorem, among other names, states that if is a random variable whose expected value is defined, and is any random variable on the same probability space, then i.e., the expected value of the conditional expected value of given is the same as the expected value of .

Law of total variance

In probability theory, the law of total variance or variance decomposition formula or conditional variance formulas or law of iterated variances also known as Eve's law, states that if and are random variables on the same probability space, and the variance of is finite, then In language perhaps better known to statisticians than to probability theorists, the two terms are the "unexplained" and the "explained" components of the variance respectively (cf. fraction of variance unexplained, explained variation).

Random variable

A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. The term 'random variable' can be misleading as it is not actually random nor a variable, but rather it is a function from possible outcomes (e.g., the possible upper sides of a flipped coin such as heads and tails ) in a sample space (e.g., the set ) to a measurable space (e.g., in which 1 corresponding to and −1 corresponding to ), often to the real numbers.

Exchangeable random variables

In statistics, an exchangeable sequence of random variables (also sometimes interchangeable) is a sequence X1, X2, X3, ... (which may be finitely or infinitely long) whose joint probability distribution does not change when the positions in the sequence in which finitely many of them appear are altered. Thus, for example the sequences both have the same joint probability distribution. It is closely related to the use of independent and identically distributed random variables in statistical models.

Complex random variable

In probability theory and statistics, complex random variables are a generalization of real-valued random variables to complex numbers, i.e. the possible values a complex random variable may take are complex numbers. Complex random variables can always be considered as pairs of real random variables: their real and imaginary parts. Therefore, the distribution of one complex random variable may be interpreted as the joint distribution of two real random variables.

Related lectures (36)

Independence and Covariance

Explores independence and covariance between random variables, discussing their implications and calculation methods.

Conditional Expectation

Covers conditional expectation, Fubini's theorem, and their applications in probability theory.

Martingale Convergence Theorem

Explores the proof of the martingale convergence theorem and the conditions for convergence to a random variable.

Probability Review

Introduces subgaussian and subexponential random variables, conditional expectation, and Orlicz norms.

Generalized Linear Models