A Simple Central Limit Theorem Proof of the Asymptotic Distribution of the Student-t Test of the Mean

Abstract

We use the Lindberg-Levy central limit theorem (CLT), Tchebychev’s inequality, Slutsky’s theorem, and general rules for limiting distributions to demonstrate sufficient conditions under which the Student-t test statistic for the mean is asymptotically standard normal. Although there exist weaker necessary and sufficient conditions under which the same asymptotic result follows (see Gine, Gotze, and Mason [1997]), our derivation is shorter and more easily accessible to general non-specialist readers. Of course, our sufficient conditions must imply the necessary and sufficient conditions given in Gine, Gotze, and Mason (1997). In non-convergent cases, our (stronger) sufficient conditions must be breached, else the (weaker) necessary and sufficient conditions in Gine, Gotze, and Mason (1997) would be satisfied, and convergence would result. Thus, in these cases, the breached assumptions of our proof allows stronger statements to be made than could be made based on necessary and sufficient conditions alone.