Law of Large Numbers - Rate of convergence

[Apteronotus]

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

ex.
if
$$lim_{n \rightarrow \infty} \frac{1}{n} \sum Z_n = \mu$$

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,

 

[mathman]

It depends very much on the distribution functions of the random variables involved.

 

[EnumaElish]

This paper has some results without proofs.

 

[statdad]

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".