Some properties of generalized Pickands constants

We study properties of generalized Pickands constants $\mathcal{H}_{\eta}$, which appear in the extreme value theory of Gaussian processes and are defined via the limit
$$ \mathcal{H}_{\eta}=\lim_{T\to\infty}\frac{\mathcal{H}_{\eta}(T)}{T}, $$
where $\mathcal{H}_{\eta}(T)=\mathbf{E}\exp(\max_{t \in[0,T]}(\sqrt{2}\,\eta(t)-\mathrm{Var}(\eta(t))))$ and $\eta(t)$ is a centered Gaussian process with stationary increments.
We give estimates of the rate of convergence of $\mathcal{H}_{\eta}(T)/T$ to $\mathcal{H}_{\eta}$ and prove that if $\eta_{(n)}(t)$ weakly converges in $C([0,\infty))$ to $\eta(t)$, then under some weak conditions, $\lim_{n\to\infty}\mathcal{H}_{\eta_{(n)}}=\mathcal{H}_{\eta}$.
As an application we prove that $\Upsilon(\alpha)=\mathcal{H}_{B_{\alpha/2}}$ is continuous on $(0,2]$, where $B_{\alpha/2}(t)$ is a fractional Brownian motion with Hurst parameter $\alpha/2$.