# almost sure convergence

• December 13th 2011, 06:53 PM
Beaky
almost sure convergence
Let $X_{1},X_{2}...$ be iid random variables with mean $\mu$ and $E(X_{1}^{4})=1$. Set $\bar{X}_{n}=(X_{1}+...+X_{n})/n$. Show $\bar{X}_{n}\to\mu$ almost surely as $n\to \infty$.

So far all I have is convergence in probability from the weak law of large numbers. I don't see what the trick is to using the expected value.
• December 13th 2011, 08:33 PM
matheagle
Re: almost sure convergence
Have you seen the Marcinkiewicz_Zygmund Law of Large Numbers?
All you need is a first moment.
• December 13th 2011, 11:50 PM
Moo
Re: almost sure convergence
But maybe his exercise is to prove it with the 4th moment !!
Which is a common proof, but the margin is too narrow to put it down. At least right now :D