Law of Large Numbers - Rate of convergence

[Apteronotus]

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

ex.
if
[tex]
lim_{n \rightarrow \infty} \frac{1}{n} \sum Z_n = \mu
[/tex]

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

2. If so, what is the value of [tex]\alpha[/tex]?

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".

 

[blue_raver22]

I can confirm that the rate of convergence of the law of large numbers is indeed an important concept in statistics. It refers to how quickly the average of a large number of independent and identically distributed random variables approaches its expected value. In other words, it measures the speed at which the sample mean approaches the population mean as the sample size increases.

To answer your first question, yes, we can say that the sum converges to \mu as n^\alpha for some \alpha\in \Re. This is because the law of large numbers states that as the sample size increases, the sample mean will approach the population mean. And according to the definition of convergence, we can say that the rate of convergence is equal to the exponent \alpha.

However, the value of \alpha may vary depending on the specific distribution and properties of the random variables. In some cases, the rate of convergence may be faster (e.g. \alpha = 1) while in others it may be slower (e.g. \alpha = 1/2). It ultimately depends on the characteristics of the data and cannot be determined without further information.

In conclusion, the rate of convergence of the law of large numbers is an important concept to understand in order to accurately interpret and analyze data. It allows us to determine how quickly our sample mean will approach the true population mean, providing valuable insights for statistical inference and decision making.