Central Limit Theorem Question

[DavidC]

I've sometimes heard people say that the Central Limit Theorem doesn't apply. I think this property of "non-applicableness" is called Statistical Fine Print, but I'm not sure.

So, two questions:

1) Is SFP a term that we use to describe situations where the CLT doesn't apply?

2) Does the CLT always apply if we were able to get an infinite sample? /images/graemlins/smile.gif

i.e. Is SFP a matter of practicality as opposed to a mathematical absolute?

[DavidC]

FWIW, I read something on the CLT, thought it's a pretty amazing piece of math, but I don't understand the section that goes into great detail on SFP. I THINK it involved integral calculations or something like that, and my stats background is only cursory, so I don't even understand what an integral is. /images/graemlins/frown.gif

[AaronBrown]

"Statistical Fine Print" is a joking reference to the fact that there are some complicated conditions that must be met before it is safe to apply the central limit theorem. There are actually many versions of the central limit theorem (more SFP), but the most common application is to compute probabilities using only the mean and standard deviation.

For practical purposes, the three most common dangers to relying on the CLT is that your sample is not big enough, you are going too far into the tails or that your individual observations are not independent. The first two disappear if you get enough observations (but "enough" may mean more observations than could possibly fit in the known universe since the beginning of time), the last never disappears, even with an infinite sample.

[DavidC]

Thanks, Aaron, and happy birthday, btw. I hope life is treating you well. /images/graemlins/smile.gif

--Dave.

The CLT states that under some very general assumptions (finite variances ...) the probability distribution of the normalized sum of n independent random variables converges against the standard normal distribution.

So, the CLT does not apply for example if the variances of the random variables are not finite or not independent.

What do you want to compute? /images/graemlins/smile.gif

[DoomSlice]

I was under the impression that there are several corrections that can be made for sample dependence, the ones that I know the most are for serially correlated observations.

[AaronBrown]

Thank you. Life is treating me better than I deserve, that's all you can ask.

[AaronBrown]

[ QUOTE ]
I was under the impression that there are several corrections that can be made for sample dependence, the ones that I know the most are for serially correlated observations.

[/ QUOTE ]
The trouble is these depend on knowing the precise nature of the serial correlation. They amount to subtracting out the known model and applying the CLT to the residuals.

As a practical matter, even small amounts of dependence can either invalidate the CLT or require unrealistically large numbers of observations.