Empirical process | Theorems in statistics | Probability theorems
In probability theory, Donsker's theorem (also known as Donsker's invariance principle, or the functional central limit theorem), named after Monroe D. Donsker, is a functional extension of the central limit theorem. Let be a sequence of independent and identically distributed (i.i.d.) random variables with mean 0 and variance 1. Let . The stochastic process is known as a random walk. Define the diffusively rescaled random walk (partial-sum process) by The central limit theorem asserts that converges in distribution to a standard Gaussian random variable as . Donsker's invariance principle extends this convergence to the whole function . More precisely, in its modern form, Donsker's invariance principle states that: As random variables taking values in the Skorokhod space , the random function converges in distribution to a standard Brownian motion as (Wikipedia).
Andrey Dymov: A functional limit theorem for the sine-process
Abstract: It is well-known that a large class of determinantal processes including the sine-process satisfies the Central Limit Theorem. For many dynamical systems satisfying the CLT the Donsker Invariance Principle also takes place. The latter states that, in some appropriate sense, traje
From playlist Probability and Statistics
Differential Equations | Application of Abel's Theorem Example 2
We give an example of applying Abel's Theorem to construct a second solution to a differential equation given one solution. www.michael-penn.net
From playlist Differential Equations
The Campbell-Baker-Hausdorff and Dynkin formula and its finite nature
In this video explain, implement and numerically validate all the nice formulas popping up from math behind the theorem of Campbell, Baker, Hausdorff and Dynkin, usually a.k.a. Baker-Campbell-Hausdorff formula. Here's the TeX and python code: https://gist.github.com/Nikolaj-K/8e9a345e4c932
From playlist Algebra
Niles Weed :Weak limits for entropic optimal transport I
CONFERENCE Recording during the thematic meeting : "Meeting in Mathematical Statistics " the December 16, 2022 at the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Jean Petit Find this video and other talks given by worldwide mathematicians on CIRM's Au
From playlist Probability and Statistics
Large deviation theory applied to climate dynamics by Freddy Bouchet
Large deviation theory in statistical physics: Recent advances and future challenges DATE: 14 August 2017 to 13 October 2017 VENUE: Madhava Lecture Hall, ICTS, Bengaluru Large deviation theory made its way into statistical physics as a mathematical framework for studying equilibrium syst
From playlist Large deviation theory in statistical physics: Recent advances and future challenges
Differential Equations | Application of Abel's Theorem Example 1
We give an example of applying Abel's Theorem to construct a second solution to a differential equation given one solution. www.michael-penn.net
From playlist Differential Equations
Hillel Furstenberg - The Abel Prize interview 2020
00:00 Congratulations 00:30 Furstenberg tells us about his childhood and his love for mathematics 03:20 Enjoying problem-solving challenges 05:44 Being an undergraduate student at Yeshiva College and his paper "on the infinitude of primes" 08:27 PhD thesis at Princeton University proving
From playlist The Abel Prize Interviews
Srinivasa Varadhan - The Abel Prize interview 2007
0:00 Abel Prize Ceremonies (Norwegian) 01:00 Interview with Skau and Raussen starts 02:30 Why so long for probability or statistics to be recognised? 04:35 Born and raised on Chennai, studied at Madras; mathematical influences 05:52 Excellent math. teacher, math. for enjoyment 07:30 Why gr
From playlist The Abel Prize Interviews
Niles Weed :Weak limits for entropic optimal transport II
CONFERENCE Recording during the thematic meeting : "Meeting in Mathematical Statistics " the December 15, 2022 at the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by worldwide mathematicians on
From playlist Probability and Statistics
11. Parametric Hypothesis Testing (cont.) and Testing Goodness of Fit
MIT 18.650 Statistics for Applications, Fall 2016 View the complete course: http://ocw.mit.edu/18-650F16 Instructor: Philippe Rigollet In this lecture, Prof. Rigollet talked about Glivenko-Cantelli Theorem (fundamental theorem of statistics), Donsker’s Theorem, and Kolmogorov-Smirnov test
From playlist MIT 18.650 Statistics for Applications, Fall 2016
Large deviations for the Wiener Sausage (Lecture 2) by Frank den Hollander
Large deviation theory in statistical physics: Recent advances and future challenges DATE: 14 August 2017 to 13 October 2017 VENUE: Madhava Lecture Hall, ICTS, Bengaluru Large deviation theory made its way into statistical physics as a mathematical framework for studying equilibrium syst
From playlist Large deviation theory in statistical physics: Recent advances and future challenges
C73 Introducing the theorem of Frobenius
The theorem of Frobenius allows us to calculate a solution around a regular singular point.
From playlist Differential Equations
In this video, I present another example of Stokes theorem, this time using it to calculate the line integral of a vector field. It is a very useful theorem that arises a lot in physics, for example in Maxwell's equations. Other Stokes Example: https://youtu.be/-fYbBSiqvUw Yet another Sto
From playlist Vector Calculus
The Fundamental Theorem of Calculus | Algebraic Calculus One | Wild Egg
In this video we lay out the Fundamental Theorem of Calculus --from the point of view of the Algebraic Calculus. This key result, presented here for the very first time (!), shows how to generalize the Fundamental Formula of the Calculus which we presented a few videos ago, incorporating t
From playlist Algebraic Calculus One
Cayley-Hamilton Theorem: General Case
Matrix Theory: We state and prove the Cayley-Hamilton Theorem over a general field F. That is, we show each square matrix with entries in F satisfies its characteristic polynomial. We consider the special cases of diagonal and companion matrices before giving the proof.
From playlist Matrix Theory
Discrete Math - 4.1.1 Divisibility
The definition and properties of divisibility with proofs of several properties. Formulas for quotient and remainder, leading into modular arithmetic. Textbook: Rosen, Discrete Mathematics and Its Applications, 7e Playlist: https://www.youtube.com/playlist?list=PLl-gb0E4MII28GykmtuBXNU
From playlist Discrete Math I (Entire Course)
Rahim Moosa: Around Jouanolou-type theorems
Abstract: In the mid-90’s, generalising a theorem of Jouanolou, Hrushovski proved that if a D-variety over the constant field C has no non-constant D-rational functions to C, then it has only finitely many D-subvarieties of codimension one. This theorem has analogues in other geometric con
From playlist Combinatorics
Calculus 1 (Stewart) Ep 22, Mean Value Theorem (Oct 28, 2021)
This is a recording of a live class for Math 1171, Calculus 1, an undergraduate course for math majors (and others) at Fairfield University, Fall 2021. The textbook is Stewart. PDF of the written notes, and a list of all episodes is at the class website. Class website: http://cstaecker.f
From playlist Math 1171 (Calculus 1) Fall 2021
Equidistribution of Unipotent Random Walks on Homogeneous spaces by Emmanuel Breuillard
PROGRAM : ERGODIC THEORY AND DYNAMICAL SYSTEMS (HYBRID) ORGANIZERS : C. S. Aravinda (TIFR-CAM, Bengaluru), Anish Ghosh (TIFR, Mumbai) and Riddhi Shah (JNU, New Delhi) DATE : 05 December 2022 to 16 December 2022 VENUE : Ramanujan Lecture Hall and Online The programme will have an emphasis
From playlist Ergodic Theory and Dynamical Systems 2022
Convolution Theorem: Fourier Transforms
Free ebook https://bookboon.com/en/partial-differential-equations-ebook Statement and proof of the convolution theorem for Fourier transforms. Such ideas are very important in the solution of partial differential equations.
From playlist Partial differential equations