**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Lecture# Martingale Convergence Theorem

Description

This lecture covers the proof of the martingale convergence theorem, demonstrating the convergence of a sequence of random variables in L2 space. The instructor explains the steps to prove the convergence of the martingale sequence and the conditions required for the convergence to a random variable M infinity. The lecture concludes with the verification of the conditional expectation of M infinity given fn, emphasizing the importance of the Martingale being closed at infinity.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

In course

Instructors (2)

Related concepts (29)

COM-417: Advanced probability and applications

In this course, various aspects of probability theory are considered. The first part is devoted to the main theorems in the field (law of large numbers, central limit theorem, concentration inequaliti

In probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to statistics and stochastic processes. The same concepts are known in more general mathematics as stochastic convergence and they formalize the idea that a sequence of essentially random or unpredictable events can sometimes be expected to settle down into a behavior that is essentially unchanging when items far enough into the sequence are studied.

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.

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.

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.

In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usually abbreviated as i.i.d., iid, or IID. IID was first defined in statistics and finds application in different fields such as data mining and signal processing. Statistics commonly deals with random samples. A random sample can be thought of as a set of objects that are chosen randomly.

Related lectures (270)

Stochastic Processes: Symmetric Random WalkFIN-415: Probability and stochastic calculus

Covers the properties of the symmetric random walk in stochastic processes.

Martingale Theory: Basics and ApplicationsMATH-467: Probabilistic methods in combinatorics

Covers the basics of Martingale theory and its applications in random variables.

Conditional expectationMATH-332: Stochastic processes

Explores the properties of conditional expectation and its extension to positive variables.

Independence and ProductsMATH-432: Probability theory

Covers independence between random variables and product measures in probability theory.

Conditional ExpectationMATH-431: Theory of stochastic calculus

Explores the properties and definition of conditional expectation in random variables.