Comments on: 275A, Notes 3: The weak and strong law of large numbers
https://terrytao.wordpress.com/2015/10/23/275a-notes-3-the-weak-and-strong-law-of-large-numbers/
Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence TaoTue, 12 Feb 2019 16:39:24 +0000
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By: Terence Tao
https://terrytao.wordpress.com/2015/10/23/275a-notes-3-the-weak-and-strong-law-of-large-numbers/#comment-512543
Tue, 12 Feb 2019 16:39:24 +0000http://terrytao.wordpress.com/?p=8486#comment-512543Yes, in these notes “ is a.s. unbounded” is short for “The sequence is a.s. unbounded”, or equivalently that is almost surely infinite. One can also replace the supremum by a limit superior if desired. (Generally speaking, the terms “bounded” and “unbounded” only apply to functions and sequences, not to scalars, where one instead uses the distinction between “finite” and “infinite”.)
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By: haonanz
https://terrytao.wordpress.com/2015/10/23/275a-notes-3-the-weak-and-strong-law-of-large-numbers/#comment-512542
Tue, 12 Feb 2019 16:26:34 +0000http://terrytao.wordpress.com/?p=8486#comment-512542Thanks for your reply. I think I have found my source of confusion. I think the statement a.s unbounded here means the is a.s unbounded as opposed to every term in the sequence is a.s unbounded. As if the later is the case (for simplicity I will denote my random variables as ). By definition of a.s unbounded, would be true for all M and n, which I can use to conclude that cannot converge to any finite value (more direct one is zero). Using the typer writer example, then one I can think about is some kind of moving train as (n,n+1/n) with positive probability in the interval and zero outside. Is my understanding here correct here? Thanks again,
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By: Terence Tao
https://terrytao.wordpress.com/2015/10/23/275a-notes-3-the-weak-and-strong-law-of-large-numbers/#comment-512348
Mon, 11 Feb 2019 02:13:30 +0000http://terrytao.wordpress.com/?p=8486#comment-512348I am not sure what the actual question is here, but you can try proving to yourself the opposite claim “A sequence of a.s. unbounded random variables cannot converge in probability to (say) zero” and see where any arguments you have in mind to justify this break down. (One can also concoct an example by modifying the “typewriter sequence”, see Example 4 of https://terrytao.wordpress.com/2010/10/02/245a-notes-4-modes-of-convergence/ .)
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By: haonanz
https://terrytao.wordpress.com/2015/10/23/275a-notes-3-the-weak-and-strong-law-of-large-numbers/#comment-512343
Sun, 10 Feb 2019 22:06:27 +0000http://terrytao.wordpress.com/?p=8486#comment-512343Hello Professor. Tao, I have question regard the result of Saint Petersburg Paradox, as in your note, we have shown converges to 1 in probability, and in Exercise 19 we can show it is almost surely unbounded. I see this as another example to show that convergence in probability is a weaker notion than a.s convergence, However I could not convince myself without going through the proof a sequence of a.s unbounded random variable can converge in probability to a finite value. Thanks,
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By: Terence Tao
https://terrytao.wordpress.com/2015/10/23/275a-notes-3-the-weak-and-strong-law-of-large-numbers/#comment-512138
Mon, 04 Feb 2019 16:47:04 +0000http://terrytao.wordpress.com/?p=8486#comment-512138 The uniform probability measure on is , not (the total measure has to equal 1 to be a probability measure). Similarly, the uniform probability measure on would be .
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By: Anonymous
https://terrytao.wordpress.com/2015/10/23/275a-notes-3-the-weak-and-strong-law-of-large-numbers/#comment-512111
Mon, 04 Feb 2019 09:56:43 +0000http://terrytao.wordpress.com/?p=8486#comment-512111 And the 1/2? Following Theorem 32 I obtain .
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By: Terence Tao
https://terrytao.wordpress.com/2015/10/23/275a-notes-3-the-weak-and-strong-law-of-large-numbers/#comment-512058
Sat, 02 Feb 2019 18:06:17 +0000http://terrytao.wordpress.com/?p=8486#comment-512058 The distribution of is symmetric around the origin, and is an even function of , so is equal to .
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By: Anonymous
https://terrytao.wordpress.com/2015/10/23/275a-notes-3-the-weak-and-strong-law-of-large-numbers/#comment-512019
Fri, 01 Feb 2019 17:46:54 +0000http://terrytao.wordpress.com/?p=8486#comment-512019I still don’t see how the limits of integration changed to [0,1].
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By: Terence Tao
https://terrytao.wordpress.com/2015/10/23/275a-notes-3-the-weak-and-strong-law-of-large-numbers/#comment-511903
Wed, 30 Jan 2019 03:56:05 +0000http://terrytao.wordpress.com/?p=8486#comment-511903See Theorem 32 of Notes 1.
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By: Anonymous
https://terrytao.wordpress.com/2015/10/23/275a-notes-3-the-weak-and-strong-law-of-large-numbers/#comment-511885
Tue, 29 Jan 2019 18:14:21 +0000http://terrytao.wordpress.com/?p=8486#comment-511885Hi Prof. Tao. Could you please clarify how you are using the change of variables formula in the proof of proposition 6?
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