On the heavy-tailedness of Student's $t$-statistic

Abstract

Let $\{X_i\}_{i\geq1}$ be an i.i.d. sequence of random variables and define, for $n\geq2$, \[T_n=\cases{n^{-1/2}\hat{\sigma}_n^{-1}S_n,\quad \hat{\sigma}_n>0,\cr 0,\quad \hat{\sigma}_n=0,}with S_n=\sum_{i=1}^nX_i, \hat{\sigma}^2_n=\frac{1}{n-1}\sum_{i=1}^n(X_i-n^{-1}S_n)^2.\] We investigate the connection between the distribution of an observation $X_i$ and finiteness of $\mathrm{E}|T_n|^r$ for $(n,r)\in \mathbb{N}_{\geq2}\times\mathbb{R}^+$. Moreover, assuming $T_n\stackrel{d}{\longrightarrow}T$, we prove that for any $r>0$, $\lim_{n\to\infty}\mathrm{E}|T_n|^r=\mathrm{E}|T|^r<\infty$, provided there is an integer $n_0$ such that $\mathrm {E}|T_{n_0}|^r$ is finite.