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

TL;DR

This paper explores the tail behavior of Student's t-statistic, linking the distribution of data points to the finiteness of moments of the statistic, and establishing conditions for the convergence of these moments.

Contribution

It provides new insights into the relationship between data distribution and the moments of Student's t-statistic, including conditions for their finiteness and convergence.

Findings

Finiteness of moments depends on the distribution of individual data points.

Under certain conditions, moments of the t-statistic converge as sample size increases.

Established criteria for the finiteness of moments for the 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.