Law of Large Numbers - Rate of convergence

Apteronotus
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What is the rate of convergence of the law of large numbers?

ex.
if
<br /> lim_{n \rightarrow \infty} \frac{1}{n} \sum Z_n = \mu<br />

1. can we say that the sum converges to \mu as n^\alpha for some \alpha\in \Re?

2. If so, what is the value of \alpha?

Thanks,
 
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It depends very much on the distribution functions of the random variables involved.
 
This paper has some results without proofs.
 
Theorems generically known as "Laws of the iterated logarithm" will give some answers. You can find discussions in probability texts (Chung, for example). A very good discussion is in the book "Approximation Theorems of Mathematical Statistics".
 

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