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

Let $\{X_i{\}}_{i\ge 1}$ be an i.i.d. sequence of random variables and define, for $n\ge 2$, $$T_n=\text{cases}{n}^{-1\mathrm{/}2}{\widehat{\sigma }}_n^{-1}{S}_n,{\widehat{\sigma }}_n>0,\text{cr}0,{\widehat{\sigma }}_n=0,with{S}_n=\sum _{i=1}^n{X}_i,{\widehat{\sigma }}_n^2=\frac{1}{n-1}\sum _{i=1}^n(X_i-n^{-1}{S}_n{)}^2\mathrm{.}$$ We investigate the connection between the distribution of an observation $X_i$ and finiteness of $\mathrm{E}\mathrm{\mid }{T}_n{\mathrm{\mid }}^r$ for $(n,r)\in {\mathbb{N}}_{\ge 2}\times {\mathbb{R}}^{+}$. Moreover, assuming $T_{n}T$, we prove that for any $r>0$, ${\lim }_{n\to \mathrm{\infty }}\mathrm{E}\mathrm{\mid }{T}_n{\mathrm{\mid }}^r=\mathrm{E}\mathrm{\mid }T{\mathrm{\mid }}^r<\mathrm{\infty }$, provided there is an integer $n_0$ such that $\mathrm{E}\mathrm{\mid }{T}_{n_0}{\mathrm{\mid }}^r$ is finite.