Lecture

Martingales and Brownian Motion: Convergence Criteria and Theorems

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In course

MATH-330: Martingales and Brownian motion

Introduction to the theory of discrete-time martingales (optional stopping and convergence theorems) and its applications (gambler's ruin, branching processes, Pólya urns...). Introduction to Brownian motion. ...

Description

This lecture delves into convergence criteria in relation to martingales, covering topics such as convergence in probability and almost sure convergence. The instructor explains the theorems of monotone convergence and dominated convergence, which allow for the interchange of limit and expectation. The lecture also introduces the concept of Cauchy criterion for almost sure convergence, providing insights into verifying the convergence of a sequence of random variables. Additionally, the instructor discusses the Cauchy criterion for martingales, highlighting its role in determining the convergence of a martingale sequence. The lecture concludes with the presentation of the first theorem of martingale convergence, emphasizing the importance of square integrable martingales.

Official source

Related concepts (32)

Convergence of random variables

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.

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.

Uniform convergence

In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions converges uniformly to a limiting function on a set as the function domain if, given any arbitrarily small positive number , a number can be found such that each of the functions differs from by no more than at every point in .

Probability

Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty. The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin.

Integral test for convergence

In mathematics, the integral test for convergence is a method used to test infinite series of monotonous terms for convergence. It was developed by Colin Maclaurin and Augustin-Louis Cauchy and is sometimes known as the Maclaurin–Cauchy test. Consider an integer N and a function f defined on the unbounded interval , on which it is monotone decreasing. Then the infinite series converges to a real number if and only if the improper integral is finite. In particular, if the integral diverges, then the series diverges as well.

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