• Bernoulli
  • Volume 17, Number 1 (2011), 276-289.

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

Fredrik Jonsson

Full-text: Open access


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.

Article information

Bernoulli, Volume 17, Number 1 (2011), 276-289.

First available in Project Euclid: 8 February 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

finiteness of moments robustness Student’s $t$-statistic $t$-distributions $t$-test


Jonsson, Fredrik. On the heavy-tailedness of Student’s $t$-statistic. Bernoulli 17 (2011), no. 1, 276--289. doi:10.3150/10-BEJ262.

Export citation


  • [1] Chistyakov, G.P. and Götze, F. (2004). Limit distributions of studentized means. Ann. Probab. 34 28–71.
  • [2] Cohn, D.L. (1980). Measure Theory. Boston, MA: Birkhäuser.
  • [3] Giné, E., Götze, F. and Mason, D.M. (1997). When is the Student t-statistic asymptotically standard normal? Ann. Probab. 25 1514–1531.
  • [4] Gut, A. (2007). Probability: A Graduate Course, Corr. 2nd printing. New York: Springer.
  • [5] Jonsson, F. (2008). Existence and convergence of moments of Student’s t-statistic. Licentiate thesis, U.U.D.M. Report 2008:18.
  • [6] Praetz, P.D. (1972). The distribution of share price changes. J. Business 45 49–55.
  • [7] Zabell, S.L. (2008). On Student’s 1908 article “The probable error of a mean”. With comments and a rejoinder by the author. J. Amer. Statist. Assoc. 481 1–20.