Law of Large Numbers - Rate of convergence

1. Feb 9, 2010

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,

2. Feb 9, 2010

mathman

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

3. Feb 24, 2010

EnumaElish

This paper has some results without proofs.

4. Feb 25, 2010

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".