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

Abstract

Let $\{X_i{\}}_{i\ge 1}$ be an i.i.d. sequence of random variables and define, for $n\ge 2$, \[T_n=\cases{n^{-1/2}\hatσ_n^{-1}S_n,\quad \hatσ_n>0,\cr 0,\quad \hatσ_n=0,}with S_n=\sum_{i=1}^nX_i, \hatσ^2_n=\frac{1}{n-1}\sum_{i=1}^n(X_i-n^{-1}S_n)^2.\] We investigate the connection between the distribution of an observation $X_i$ and finiteness of $\mathrm{E}|T_n{|}^r$ for $(n,r)\in {\mathbb{N}}_{\ge 2}\times {\mathbb{R}}^{+}$. Moreover, assuming $T_n\stackrel{d}{⟶}T$, 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.

Published in at http://dx.doi.org/10.3150/10-BEJ262 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)