Article proofs | Statistical randomness

Proofs of convergence of random variables

This article is supplemental for “Convergence of random variables” and provides proofs for selected results. Several results will be established using the portmanteau lemma: A sequence {Xn} converges in distribution to X if and only if any of the following conditions are met: 1. * E[f(Xn)] → E[f(X)] for all bounded, continuous functions f; 2. * E[f(Xn)] → E[f(X)] for all bounded, Lipschitz functions f; 3. * limsup{Pr(Xn ∈ C)} ≤ Pr(X ∈ C) for all closed sets C; (Wikipedia).

Video thumbnail

How to Prove a Sequence with Two Components Converges

How to Prove a Sequence with Two Components Converges If you enjoyed this video please consider liking, sharing, and subscribing. You can also help support my channel by becoming a member https://www.youtube.com/channel/UCr7lmzIk63PZnBw3bezl-Mg/join Thank you:)

From playlist Advanced Calculus

Video thumbnail

Calculus: How Convergence Explains The Limit

The limit definition uses the idea of convergence twice (in two slightly different ways). Once the of convergence is grasped, the limit concept becomes easy, even trivial. This clip explains convergence and shows how it can be used to under the limit.

From playlist Summer of Math Exposition Youtube Videos

Video thumbnail

Find the Interval of Convergence

How to find the interval of convergence for a power series using the root test.

From playlist Convergence (Calculus)

Video thumbnail

Proof: Limit Law for Difference of Convergent Series | Real Analysis

We prove the limit law for the difference of convergent series. If two series converge to a and b, then the series whose terms are the differences of the terms of the original two series is the difference of the limits: a-b. We'll prove this using the limit law for the difference of conver

From playlist Real Analysis

Video thumbnail

The Law of Large Numbers - Explained

The law of large numbers is one of the most intuitive ideas in statistics, however, often the strong and weak versions of the law can be difficult to understand. In this video, I breakdown what the definitions of both laws mean and use this as a way to introduce the concepts of convergence

From playlist Summer of Math Exposition 2 videos

Video thumbnail

L18.6 Convergence in Probability

MIT RES.6-012 Introduction to Probability, Spring 2018 View the complete course: https://ocw.mit.edu/RES-6-012S18 Instructor: John Tsitsiklis License: Creative Commons BY-NC-SA More information at https://ocw.mit.edu/terms More courses at https://ocw.mit.edu

From playlist MIT RES.6-012 Introduction to Probability, Spring 2018

Video thumbnail

Infinity Paradox -- Riemann series theorem

Absolute Convergence versus Conditional Convergence

From playlist Physics

Video thumbnail

Absolute Convergence, Conditional Convergence, and Divergence

This calculus video tutorial provides a basic introduction into absolute convergence, conditional convergence, and divergence. If the absolute value of the series convergences, then the original series will converge based on the absolute convergence test. If the absolute value of the ser

From playlist New Calculus Video Playlist

Video thumbnail

Proof: The Limit of a Sequence is Unique | Real Analysis

A convergent sequence converges to exactly one limit. That is, the limit of a sequence is unique. We'll prove this by contradiction in today's real analysis video lesson. We assume our convergent sequence converges to a and b, and that they are distinct, as in the limit is not unique. We t

From playlist Real Analysis

Video thumbnail

3. Law of Large Numbers, Convergence

MIT 6.262 Discrete Stochastic Processes, Spring 2011 View the complete course: http://ocw.mit.edu/6-262S11 Instructor: Robert Gallager License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu

From playlist MIT 6.262 Discrete Stochastic Processes, Spring 2011

Video thumbnail

Benjamin Weiss: Poisson-generic points

CIRM VIRTUAL CONFERENCE Recorded during the meeting "​ Diophantine Problems, Determinism and Randomness" the November 25, 2020 by the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by worldwide

From playlist Virtual Conference

Video thumbnail

Lecture for Workshop on Random Matrices and Random Systems - Balint Virag

Balint Virag Univ Toronto April 1, 2014 For more videos, visit http://video.ias.edu

From playlist Mathematics

Video thumbnail

4.5.1 Expectation: Video

MIT 6.042J Mathematics for Computer Science, Spring 2015 View the complete course: http://ocw.mit.edu/6-042JS15 Instructor: Albert R. Meyer License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu

From playlist MIT 6.042J Mathematics for Computer Science, Spring 2015

Video thumbnail

Singular Learning Theory - Seminar 13 - Main Theorem 2

This seminar series is an introduction to Watanabe's Singular Learning Theory, a theory about algebraic geometry and statistical learning theory. In this seminar Edmund Lau proves Main Theorem 2 from the "grey book". The webpage for this seminar is https://metauni.org/slt/ You can join t

From playlist Singular Learning Theory

Video thumbnail

Brian Rider: Operator limits of beta ensembles - Lecture 2

Abstract: Random matrix theory is an asymptotic spectral theory. For a given ensemble of n by n matrices, one aims to proves limit theorems for the eigenvalues as the dimension tends to infinity. One of the more remarkable aspects of the subject is that it has introduced important new poin

From playlist Analysis and its Applications

Video thumbnail

Rumours, consensus and epidemics on networks (Lecture 1) by A Ganesh

PROGRAM : ADVANCES IN APPLIED PROBABILITY ORGANIZERS : Vivek Borkar, Sandeep Juneja, Kavita Ramanan, Devavrat Shah and Piyush Srivastava DATE & TIME : 05 August 2019 to 17 August 2019 VENUE : Ramanujan Lecture Hall, ICTS Bangalore Applied probability has seen a revolutionary growth in r

From playlist Advances in Applied Probability 2019

Video thumbnail

Introduction to Continuous Combinatorics II: semantic limits - Leonardo Coregliano

Computer Science/Discrete Mathematics Seminar II Topic: Introduction to Continuous Combinatorics II: semantic limits Speaker: Leonardo Coregliano Affiliation: Member, School of Mathematics Date: November 09, 2021 The field of continuous combinatorics studies large (dense) combinatorial s

From playlist Mathematics

Video thumbnail

4. Poisson (the Perfect Arrival Process)

MIT 6.262 Discrete Stochastic Processes, Spring 2011 View the complete course: http://ocw.mit.edu/6-262S11 Instructor: Robert Gallager License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu

From playlist MIT 6.262 Discrete Stochastic Processes, Spring 2011

Video thumbnail

S18.1 Convergence in Probability of the Sum of Two Random Variables

MIT RES.6-012 Introduction to Probability, Spring 2018 View the complete course: https://ocw.mit.edu/RES-6-012S18 Instructor: John Tsitsiklis License: Creative Commons BY-NC-SA More information at https://ocw.mit.edu/terms More courses at https://ocw.mit.edu

From playlist MIT RES.6-012 Introduction to Probability, Spring 2018

Video thumbnail

11. Renewals: Strong Law and Rewards

MIT 6.262 Discrete Stochastic Processes, Spring 2011 View the complete course: http://ocw.mit.edu/6-262S11 Instructor: Robert Gallager License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu

From playlist MIT 6.262 Discrete Stochastic Processes, Spring 2011

Related pages

Bounded function | Borel–Cantelli lemma | Convergence of random variables | Ball (mathematics) | Indicator function | Cumulative distribution function | Continuous function | Closed set