In this thesis, we study the Central Limit Theorem. We examine the differences between independent and pairwise independent random variables, where in both cases the variables are identically distributed and have finite variance. The aim of the thesis is to demonstrate a counterexample that shows that taking a sequence of only pairwise independent random variables, which are identically distributed, can lead to incorrect interpretation of the Central Limit Theorem. Specifically, we will construct a sequence of pairwise independent random variables and then show that the standardized mean does not converge to the standard normal distribution, as the sample size tends to the infinity. In justifying the counterexample, we will need many advanced tools such as expected value and variance of random vectors, characteristic functions, multinomial distribution, chi-square distribution, convergence in distribution, and conditional expected value.
|
