We prove the Berry-Esseen bound for the Student t-statistic. Under the assumption of a third moment this bound coincides (up to an absolute constant) with the classical Berry-Esseen bound for the mean. In general the distribution of the Student statistic converges to the standard normal distribution function at least as fast as the distribution of the mean, and sometimes faster. For example, rates of convergence can be proved if the underlying distribution is in the domain of attraction of the normal law.
"The Berry-Esseen bound for student's statistic." Ann. Probab. 24 (1) 491 - 503, January 1996. https://doi.org/10.1214/aop/1042644728