  Particularly in genomics, but also in other fields, it has become commonplace
to undertake highly multiple Student's $t$-tests based on relatively small
sample sizes. The literature on this topic is continually expanding, but the
main approaches used to control the family-wise error rate and false discovery
rate are still based on the assumption that the tests are independent. The
independence condition is known to be false at the level of the joint
distributions of the test statistics, but that does not necessarily mean, for
the small significance levels involved in highly multiple hypothesis testing,
that the assumption leads to major errors. In this paper, we give conditions
under which the assumption of independence is valid. Specifically, we derive a
strong approximation that closely links the level exceedences of a dependent
``studentized process'' to those of a process of independent random variables.
Via this connection, it can be seen that in high-dimensional, low sample-size
cases, provided the sample size diverges faster than the logarithm of the
number of tests, the assumption of independent $t$-tests is often justified.
