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

7 Pages Posted: 3 Feb 2019

See all articles by Timothy Falcon Crack

Timothy Falcon Crack

University of Otago - Department of Accountancy and Finance

Lynn McAlevey

University of Otago - Department of Accountancy and Finance

Anindya Sen

University of Otago - Department of Accountancy and Finance

Date Written: January 22, 2019

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.

Keywords: Student-t, Central Limit Theorem, Tchebychev, Asymptotic Normality

JEL Classification: B23, C12, C46

Suggested Citation

Crack, Timothy Falcon and McAlevey, Lynn and Sen, Anindya, A Simple Central Limit Theorem Proof of the Asymptotic Distribution of the Student-t Test of the Mean (January 22, 2019). Available at SSRN: https://ssrn.com/abstract=3320243 or http://dx.doi.org/10.2139/ssrn.3320243

Timothy Falcon Crack (Contact Author)

University of Otago - Department of Accountancy and Finance ( email )

Dunedin
New Zealand

Lynn McAlevey

University of Otago - Department of Accountancy and Finance ( email )

Dunedin
New Zealand

Anindya Sen

University of Otago - Department of Accountancy and Finance ( email )

PO Box 56
Dunedin, 9054
New Zealand

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