sLLN vs. wLLN
I would love to have an example for strong vs. weak law of large numbers:
* the strong law requires iid distribution and such that as such.
* the weak law requires and but does not have to be iid.
What would be an example of a non-iid function that fails the strong law but holds up with the weak law?
DO you really want examples of these?
I have a slew of them.
See the st petersburg game.
Your conditions are incorrect.
The point is, not iid, but whether the mean exists.
Let your sequence be iid with density
You can obtain a WLLN but not a SLLN, due to the Borel-Cantelli lemma.
1) correcting my conditions, would have missed that;
2) reference to St. Petersburg game - not sure though where to find it (are you talking about the board game? I checked the arcade and didn't see a St. Petersburg game in there) - can you send me in the right direction? I really would like examples.
3) saving me a bit of anxiety on my last exam: your suggestion (in response to one of my posts a little while ago) to use mgf's for getting moments stuck in my head and turned out to be really, really useful. ;-)
google st petersburg paradox
It's 300th anniversary is coming up
The point to that game was trying to find a fair price when the rvs were
a transformed geometric...
Here the first moment is infinite but barely.
These are the type of rvs I study.
Feller Volume 1 page 251-253 has a weak law solution for the st pete game
BUT that solution is lame.
Klass and Teicher proved in 78 that the almost sure behaviour is such that...
That's base 2, by the way.
and almost surely
The Weak Law proved in Feller is that...
You can see a Stong Law fails via Borel-Cantelli.
An easier case is what I mentioned last week.
HOWEVER, if you observe a weighted verson of these sums....
you can obtain a strong law
What's weird about that comment, was that I hesitated in making it.
Originally Posted by Statistik
The post was about using the density and not the MGF to derive the fourth moment. In the past I've received criticism for answering a question with a different technique so I thought about it several times before writing my response.
In any case, I've probably stayed past my expectations (pun included).
I never had planned to stay this long.
It's been a year and I should be doing more of my own research anyhow.
I'll appreciate feedback / questions / suggestions of any sort anytime. It's about learning (right? ;-) ), so getting to think outside of the outlined box is good and probably more realistic in the long run anyway.
Re: sLLN vs. wLLN
matheagle, could you please tell me where I can find the math you mentioned? If you kindly write the solution briefly, that will be in fact great!
Originally Posted by matheagle