When is the student t-statistic asymptotically standard normal?

Abstract

Let X, Xi, i ∈ ℕ, be independent, identically distributed random variables. It is shown that the Student t-statistic based upon the sample {Xi}ni=1 is asymptotically N(0, 1) if and only if X is in the domain of attraction of the normal law. It is also shown that, for any X, if the self-normalized sums Un := ∑ni=1 Xi/(∑ni=1 X2i)1/2 n ∈ ℕ, are stochastically bounded then they are uniformly subgaussian that is, supn double-struck E exp(λU2n < ∝ for some λ > 0. Student t-statistic, self-normalized sums, domains of attraction, convergence of moments.