  We prove new lower bounds for the upper tail probabilities of suprema of
Gaussian processes. Unlike many existing bounds, our results are not
asymptotic, but supply strong information when one is only a little into the
upper tail. We present an extended application to a Gaussian version of a
random process studied by Halasz. This leads to much improved lower bound
results for the sum of a random multiplicative function. We further illustrate
our methods by improving lower bounds for some classical constants from extreme
value theory, the Pickands constants $H_{\alpha}$, as $\alpha\rightarrow 0$.
