For appropriate Gaussian processes, as a corollary of the majorizing measure theorem, Michel Talagrand (1987) proved that the event that the supremum is significantly larger than its expectation can be covered by a set of half-spaces whose sum of measures is small. We prove a conjecture of Talagrand that is the analog of this result in the Bernoulli-$p$ setting, and answer a question of Talagrand on the analogous result for general positive empirical processes.

翻译：对于适当的高斯过程，作为主测度定理的一个推论，Michel Talagrand（1987）证明了上确界显著大于其期望的事件可以被一组测度之和很小的半空间所覆盖。我们证明了Talagrand的一个猜想，该猜想是这一结果在伯努利-p情形下的类比，并回答了Talagrand关于一般正经验过程类似结果的问题。