Centralni limitni izrek in paroma neodvisne slučajne spremenljivke ; Central limit theorem and pairwise independent random variables

V diplomski nalogi preučujemo centralni limitni izrek. Pogledamo si razlike med neodvisnimi in paroma neodvisnimi slučajnimi spremenljivkami, kjer so v obeh primerih spremenljivke enako porazdeljene in imajo končno varianco. Cilj diplomske naloge je prikaz protiprimera, ki pokaže, da lahko, ko vzamemo zaporedje zgolj paroma neodvisnih slučajnih spremenljivk, ki so enako porazdeljene, centralni limitni izrek ne velja več. Eksplicitno bomo zgradili zaporedje paroma neodvisnih slučajnih spremenljivk in potem pokazali, da standardizirana povprečja ne konvergirajo k standardni normalni porazdelitvi. Pri utemeljevanju protiprimera bomo potrebovali veliko naprednih orodij, kot so: pričakovana vrednost in varianca slučajnih vektorjev, karakteristične funkcije, multinomska porazdelitev, porazdelitev hi kvadrat, konvergenca v porazdelitvi in pogojna pričakovana vrednost. ; 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.