Limit distributions of Studentized means

Abstract

Let $X,X_j,j\in \mathbb{N}$, be independent, identically distributed random variables with probability distribution F. It is shown that Student's statistic of the sample $\{X_j\}_{j=1}^n$ has a limit distribution G such that $G(\{-1,1\})\ne 1$, if and only if: (1) $X$ is in the domain of attraction of a stable law with some exponent $0<\alpha\le 2$; (2) $\E X=0$ if $1<\alpha\le 2;$ (3) if $\alpha=1$, then X is in the domain of attraction of Cauchy's law and Feller's condition holds: $\lim_{n\to\infty}n\E\sin(X/a_n)$ exists and is finite, where $a_n$ is the infimum of all $x>0$ such that $nx^{-2}(1+\int_{(-x,x)} y^2 F \{dy\})\le 1$. If $G(\{-1,1\})=1$, then Student's statistic of the sample $\{X_j\}_{j=1}^n$ has a limit distribution if and only if $\P(|X|>x),x>0$, is a slowly varying function at $+\infty$.