**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: Proof and Stopping Time

Description

This lecture covers the proof of the martingale convergence theorem, focusing on the key ingredient of maximal inequality for square-integrable martingales. The instructor explains the concept of stopping time and its application in the context of martingales, emphasizing the control over the behavior of the process. Various mathematical expressions and inequalities are derived to illustrate the convergence properties of martingales.

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

Related concepts (26)

Instructors (2)

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

Mathematical proof

A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation".

Proof (truth)

A proof is sufficient evidence or a sufficient argument for the truth of a proposition. The concept applies in a variety of disciplines, with both the nature of the evidence or justification and the criteria for sufficiency being area-dependent. In the area of oral and written communication such as conversation, dialog, rhetoric, etc., a proof is a persuasive perlocutionary speech act, which demonstrates the truth of a proposition.

Constructive proof

In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also known as an existence proof or pure existence theorem), which proves the existence of a particular kind of object without providing an example. For avoiding confusion with the stronger concept that follows, such a constructive proof is sometimes called an effective proof.

Stopping time

In probability theory, in particular in the study of stochastic processes, a stopping time (also Markov time, Markov moment, optional stopping time or optional time) is a specific type of “random time”: a random variable whose value is interpreted as the time at which a given stochastic process exhibits a certain behavior of interest. A stopping time is often defined by a stopping rule, a mechanism for deciding whether to continue or stop a process on the basis of the present position and past events, and which will almost always lead to a decision to stop at some finite time.

Proof by exhaustion

Proof by exhaustion, also known as proof by cases, proof by case analysis, complete induction or the brute force method, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases or sets of equivalent cases, and where each type of case is checked to see if the proposition in question holds. This is a method of direct proof. A proof by exhaustion typically contains two stages: A proof that the set of cases is exhaustive; i.e.

Related lectures (316)

Convergence CriteriaMATH-101(g): Analysis I

Covers the convergence criteria for sequences, including operations on limits and sequences defined by recurrence.

Proofs and Logic: Introduction

Introduces logic, proofs, sets, functions, and algorithms in mathematics and computer science.

Distributions and DerivativesMATH-502: Distribution and interpolation spaces

Covers distributions, derivatives, convergence, and continuity criteria in function spaces.

Sobolev Spaces in Higher DimensionsMATH-502: Distribution and interpolation spaces

Explores Sobolev spaces in higher dimensions, discussing derivatives, properties, and challenges with continuity.

Harmonic Forms: Main TheoremMATH-410: Riemann surfaces

Explores harmonic forms on Riemann surfaces and the uniqueness of solutions to harmonic equations.