WHEN IS THE STUDENT T-STATISTIC ASYMPTOTICALLY STANDARD NORMAL

Let X, X-i, i is an element of N, be independent, identically distribu ted random variables. It is shown that the Student t-statistic based u pon the sample {X-i}(i less than or equal to 1)(n) is asymptotically N (0, 1) if and only if X is in the domain of attraction of the normal l aw. It is also shown that, for any X, if the self-normalized sums U-n := Sigma(i=1)(n) X-i/(Sigma(i=1)(n) X-i(2))(1/2), n is an element of N , are stochastically bounded then they are uniformly subgaussian that is, sup(n) E exp(lambda U-n(2)) < infinity for some lambda > 0.