Proof of strong law of large number

[Darkzo-n]

Hi, i had recently come across the proof of the strong law of large number.
Inside the proof , it assumed the random variable have a finite fourth moment. may i know y is the assumption necessary and why the 4th? why not 2nd or 3rd or 5th?

 

[chiro]

Hey Darkzo-n and welcome to the forums.

For this problem, the kurtosis is defined as the expectation of [(X - mu)/sigma]^4.

If the distribution has an infinite mean in either direction, then the kurtosis will not exist. If the mean is zero, then X - mu will also be zero as well so that's not an issue.

The gaurantee of a mean of some sort is what makes the Strong law have its power because things converge to some mean, and without this finite quality, the convergence can not take place and the thing won't work.