The central limit theorem under simple random sampling

Proof of the central limit theorem for the sample mean can be obtained in the case of independent and identically distributed random variables through the moment generating function under some simplifying assumptions. The case for the central limit theorem for the sample mean from finite populations under simple random sample without replacement, the parallel to the simplest case in the standard framework, is not as simple. Most sampling a textbooks avoid any technical discussion of the finite population central limit theorem so that there is little appreciation for the conditions that underlay the theorem. Most often the theorem is illustrated with a simulation study; but this can only hint at the technicalities of the theorem. Now, new software has been developed that allows for automatic calculation of moments and cumulants of estimators used in survey sampling, as well as automatic derivation of unbiased or consistent estimators. This software is used to examine the assumptions behind the applications of central limit theorem under simple random sampling without replacement. Moreover, the demonstration of the theorem through this software relies on the cumulant generating function thus providing a parallel to the demonstration of the theorem in the standard case.