Terence Tao's Central Limit Theorem Proof, Double Factorials (!!) and the Moment Method

An animated video version of the moment method proof of the central limit theorem, which I first read about in Terrence Tao's book “Topics in Random Matrix Theory” (see link to book below). The proof features the double factorial sequence as the bridge linking sums of random variables to Gaussians. According to Wikipedia, this proof method was originally discovered by Chebyshev in the 1800s. https://en.wikipedia.org/wiki/Method_....

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FOLLOW UP QUESTIONS
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Possible questions to answer in a follow up videos. Let me know in the comments if you would want to see any of these!

Q: Why is this in a random matrix book? A: There is a similar proof for something called the semi-circle law for random matrices. If you understand this proof, you can understand that one quite easily!

Q: Why is a sum of Gaussian’s still a Gaussian? A: There is a cute way to see this using only moments! It is a cute counting argument with pair partitions which are labeled with some extra labels.

Q: How is this related to the characteristic function/Fourier transform? A: The moments can be added together in a generating series to get things like the characteristic function .

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LINKS
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Link to Terence Tao’s book (from his terrytao blog). The moment method proof is section 2.2.3 starting on page 106.
https://terrytao.files.wordpress.com/...

Short pdf that includes the main proof ideas
https://www.cs.toronto.edu/~yuvalf/CL....

3Blue1Brown Central Limit Theorem playlist:
   • Central limit theorem  

But what is the Central Limit Theorem?
   • But what is the Central Limit Theorem?  

Why π is in the normal distribution (beyond integral tricks)
   • Why π is in the normal distribution (...  

Convolutions | Why X+Y in probability is a beautiful mess
   • Convolutions | Why X+Y in probability...  

Why is the "central limit" a normal distribution?
   • A pretty reason why Gaussian + Gaussi...  

Image of Terence Tao is a public domain image from wikipedia: https://commons.wikimedia.org/wiki/Fi...

A note I referenced on fair use of book covers https://psu.libanswers.com/faq/336502

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Info on how the video was made
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All animations were made in manim https://www.manim.community/ and I also used manim-voiceover for the voiceover https://voiceover.manim.community/en/.... Unfortunately, this took a really long time to do! One way I am hoping to make things a bit faster might be to record it as a slideshow using manim-slides https://www.manim.community/plugin/ma... ... I might try to do this on my next video so it doesn't take quite so long to record everything!

All my manim code is available at https://github.com/mcnica89/manim if you want to see it.

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Timestamps
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0:00 Terence Tao
0:38 What is the CLT
2:40 But why?
3:30 Double factorials
4:30 Gaussian moments
5:52 Moments of sums
6:29 Pair partitions
9:00 Two main results
10:08 Part 1-Gaussian moments
13:08 Solving the recurrence relation
15:35 Part 2-Moments of sums
17:17 k equals 1
18:00 k equals 2
21:47 k equals 3
25:42 k equals 4
29:41 Recap and overview


#3blue1brown #some #some3 #probability #Gaussian #math #integration #integrationbyparts #combinatorics