sLLN vs. wLLN

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March 4th 2010, 08:08 AM
Statistik

sLLN vs. wLLN

Hi,

I would love to have an example for strong vs. weak law of large numbers:

* the strong law requires iid distribution and $EX_i = \mu$ such that $\overline {X_n} \to \mu$ as such.
* the weak law requires $EX_i \to \mu$ and $VarX_i \to 0$ 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?

Thanks!
March 4th 2010, 10:52 PM
matheagle

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

$f_X(x)=x^{-2}I(x>1)$ .

You can obtain a WLLN but not a SLLN, due to the Borel-Cantelli lemma.
March 8th 2010, 02:46 PM
Statistik

Examples

Matheagle,

Thanks for
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. ;-)
March 8th 2010, 10:55 PM
matheagle

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...

$P(X=2^n)=2^{-n}$ n=1,2,3...

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...

$\liminf_{n\to\infty}{\sum_{k=1}^nX_k\over {\rm n log_2 n}}=1$ almost surely

That's base 2, by the way.

and $\limsup_{n\to\infty}{\sum_{k=1}^nX_k\over {\rm n log_2 n}}=\infty$ almost surely

The Weak Law proved in Feller is that...

${\sum_{k=1}^nX_k\over {\rm n log_2 n}}\buildrel P\over\to 1$

You can see a Stong Law fails via Borel-Cantelli.

An easier case is what I mentioned last week.

$f(x)=x^{-2}I(x>1)$

HOWEVER, if you observe a weighted verson of these sums....
http://www.sciencedirect.com/science...7c34348333de7a
you can obtain a strong law
March 8th 2010, 11:57 PM
matheagle

Quote:

Originally Posted by Statistik

Matheagle,

Thanks for

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. ;-)

What's weird about that comment, was that I hesitated in making it.
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.
March 9th 2010, 08:19 AM
Statistik

mgf's

Hey,
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.
January 31st 2013, 01:21 AM
BMW

Re: sLLN vs. wLLN

Quote:

Originally Posted by matheagle

Let your sequence be iid with density

$f_X(x)=x^{-2}I(x>1)$ .

You can obtain a WLLN but not a SLLN, due to the Borel-Cantelli lemma.

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!