LetT be any set, and let (X t )t ∈ T be a separable gaussian process with mean zero onT. Assume thatX is almost surely bounded, and let $$\mathop {N = \sup }\limits_{ t \in X} {\text{|}}X_t {\text{|}}$$ . $$\sigma = \mathop {\sup }\limits_{t \in T} (EX_t^2 )^{1/2}$$ , and let $$\tau = \sup \{ \lambda \geqq 0;P\{ Nτ/σ 2. We prove that $$E\left( {\exp \left( {\frac{1}{{2\sigma ^2 }}N^2 - \beta N} \right)} \right)< + \infty .$$ Examples are given to show that this result cannot be readily improved. When τ=0, we also show that the distribution ofN has a continuous density with respect to Lebesgue measure.