In order to emphasize the theoretic value of Taylor series we have to see what is the difference from Newtonian Mechanics (NM) and Special Relativity (SR) of Einstein. In SR the kinetic energy of a particle with mass m and velocity v is:

K=\frac {mc^2}{\sqrt {1-{\frac {{\upsilon}^{2}}{{c}^{2}}}}}-m{c}^{2}

or

K=m*c^2/sqrt(1-v^2/c^2)-m*c^2

By taking the Taylor series expansion around v=0 we have:

K=\frac{m}{2}\,{\upsilon}^{2}+\frac {3m}{8{c}^{2}}{\upsilon}^{4}+\frac {5m}{16c^4}\,{\upsilon}^{6}+O \left(\upsilon^8 \right)

or

K=1/2*m*v^2+(3*m/(8*c^2))*v^4+(5*m/(16*c^4))*v^6+O(v^8)

So the term 1/2*m*v^2 is Newton' s kinetic energy when v

This is not "the" proof, but just "a" proof. If you think it has flaws, go for another one, e.g. http://link.springer.com/article/10.1007%2FBF01240790

http://www.cs.toronto.edu/~yuvalf/CLT.pdf

Unlike the moment-generating function (or Laplace transform), the characteristic function (or Fourier transform) always exists, regardless of the inexistence of higher moments.

As I am looking the details I think that the problem is not in the proof but in the use of it. What happens if we have a Cauchy distribution for example? The CLT cannot be applied but almost nobody checks such a possibility about his/her data. That' s the reason why I would like to exist a more general Probability Density Limit Theorem.

What you are looking for is called a `stable distribution'. In the language of stable distributions, what the CLT says is that distributions with finite variance are in the `domain of attraction' of the normal. Without finite variance, the limit must be a(nother) stable distribution.

For instance, the Cauchy distribution is stable.

http://en.wikipedia.org/wiki/Stable_distribution

@Pedro, I agree with you, but I'd like to have a unified approach to this problem. Recently distributions with 'fat tails' have gained increased respect, but our theoretical 'apparatus' is not complete yet.

This issue is so interesting! I will keep my eyes on this topic! Good luck , dear Demetris

I think that this wiki:

http://en.wikipedia.org/wiki/Stable_distribution

last updated on 19 April 2013 at 16:42.

is more general.

So, if we have finite variance, then CLT holds.

If we have not finite variance ( heavy-tailed distributions), then we can find an a stable distribution as the attracting distribution, instead of the normal one.

But, in practise I do not see many researchers to worry about the existence or not of finite moments (mean, variance etc).

Can anyone provide us with resources about confidence tests for investigating the existence or not of 2nd and higher moments from a given data set?

Hi Demitris, you must not confuse the theroretical moments with empirical ones. You can calculate sample means, variances and so on from every given sample.

I had found some years ago an article that was making a kind of test for the existence of 2,3,... moments for a given sample, but it was not published. Are there any such articles that have been published till now?

.

On the existence of moments - With an application to German stock returns

Ralf Runde and Axel Scheffner

ABSTRACT:

Stock returns are often modeled as having infinite second or fourth moments, with consequences for test statistics which have not yet been fully explored. Conclusions on the existence of moments are usually drawn from a generalized Pareto or simple Pareto tail index estimate. In a recent study McCulloch (1997) demonstrated that this estimator indicates distributions with even finite fourth moments, although the samples were drawn from infinite-variance stable laws, which points out the doubtful role of the tail index estimate as evidence for the finiteness of moments. Based on an fQ-System for continuous unimodal distributions, introduced by Scheffner (1998), we derive an alternative condition for the existence of moments. An estimation algorithm for the fQ-parameters is proposed and an application to the 30 most busy German stocks shows that daily returns can be modeled as being at least approximately fQ-distributed with finite second moments.

https://www.researchgate.net/publication/2626493_On_the_existence_of_moments_-_With_an_application_to_German_stock_returns

A Simple Nonparametric Test for the Existence of Finite Moments

Igor Fedotenkov

Abstract:

This paper proposes a simple, fast and direct nonparametric test to verify if a sample is drawn from a distribution with a finite first moment. The method can also be applied to test for the existence of finite moments of another order by taking the sample to the corresponding power. The test is based on the difference in the asymptotic behaviour of the arithmetic mean between cases when the underlying probability function either has or does not have a finite first moment. Test consistency is proved; then, test performance is illustrated with Monte-Carlo simulations and a practical application for the S&P500 index.

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2202269

Does the k-th moment exist ?

Hitomi, K. and Y. Nishiyama

Abstract:

Most asymptotic distribution theory used in econometric

research relies on moment conditions which

carefully control outlier occurrences. It is not unusual

in time series analysis to see conditions of the type

let all required moments exist. However, in financial

and commodity market time series the extent of outlier

activity casts doubt on the suitability of such generic

moment conditions. Mandelbrot (1963) provided

suggestive evidence that even second moments may

not exist for this type of data, and he proposed stable

distributions with infinite variance as an alternative to

finite-variance statistical models. Subsequent research

has generally reached the conclusion that second

moments of most datasets appear to be finite.

In many empirical works in finance, the values of

kurtosis and skewness were statistically tested even

though the existence of higher-order, especially fourth

or sixth, moments has been studied less extensively.

http://www.mssanz.org.au/MODSIM07/papers/16_s9/Doesk-thMomen_s9_Hitomi_.pdf

among many, many, many others !

.

Article On the existence of moments - With an application to German ...

@Clerot, thank you. I think you "sent me to the library" in order to read:)

I' ll come back later...

I think a very good summary is here:

https://en.wikipedia.org/wiki/Power_law

Power laws although attractive need further foundation.

Almost all suggested tests are graphical, this is not satisfactory.

Anyway, here you can find useful codes:

http://tuvalu.santafe.edu/~aaronc/powerlaws/

Of course it would be nice researchers to study their data with those tools, but I do not see it happening so often.

As for the Central Limit Theorem, by the view of power laws, it seems to be the exception in nature.

I'm just a graduation student of statistics(so forgive me if I say something stupid), but i am learning the CLT with a very good and theoretic professor and he always say that it is VERY important the existence of the first and second moment (finite ones) because the proof of the CLT depends of a Taylor expansion of order 2.

It is very important for you to worry about these details in the use of such an important theorem.

Recently I was in presentation of research works and I asked one researcher:

"Have you proven that second and third moments exist for your data".

He looked me a little strange... So many people are doing work with data moments without having proven first their existence...