Bootstrapping the Student t-Statistic

Abstract

Let $X_1\ldots,X_n, n \geq 1$, be independent, identically distributed random variables and consider the Student $t$-statistic $T_n$ based upon these random variables. Giné, Götze and Mason (1997) proved that $T_n$ converges in distribution to a standard normal random variable if and only if $X$ is in the domain of attraction of a normal random variable and$EX = 0$. We shall show that roughly the same holds true for the bootstrapped Student $t$- statistic $T_n^*$. In the process we shall disclose all the possible subsequential limiting laws of $T_^*n$. The proofs introduce a number of amusing tricks that may be of independent interest.