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

Abstract

Let $\{X_i\}_{i≥1}$ be an i.i.d. sequence of random variables and define, for $n≥2$, $$T_{n}=\cases{n^{-1/2}\hat{\sigma}_{n}^{-1}S_{n}, & $\quad \hat{\sigma}_{n}>0,$ \cr 0, & $\quad \hat{\sigma}_{n}=0,$} \qquad\mbox{with }S_{n}=\sum_{i=1}^{n}X_{i}, \hat{\sigma}^{2}_{n}=\frac{1}{n-1}\sum_{i=1}^{n}(X_{i}-n^{-1}S_{n})^{2}.$$ We investigate the connection between the distribution of an observation $X_i$ and finiteness of $\mathrm{E}|T_n|^r$ for $(n,r)∈ℕ_{≥2}×ℝ^+$. Moreover, assuming $T_{n}\stackrel {d}{\longrightarrow }T$, we prove that for any $r>0, \lim _{n→∞}\mathrm{E}|T_n|^r=\mathrm{E}|T|^r<∞$, provided there is an integer $n_0$ such that $\mathrm{E}|T_{n_0}|^r$ is finite.