Let $\xi(t)$ be a standard locally stationary Gaussian process with covariance function $1-r(t,t+s)\sim C(t)|s|^\alpha$ as $s\to0$, with $0<\alpha\leq 2$ and $C(t)$ a positive bounded continuous function. We are interested in the exceedance probabilities of $\xi(t)$ with a random standard deviation $\eta(t)=\eta-\zeta t^\beta$, where $\eta$ and $\zeta$ are non-negative bounded random variables. We investigate the asymptotic behavior of the extreme values of the process $\xi(t)\eta(t)$ under some specific conditions which depends on the relation between $\alpha$ and $\beta$.
"Extremes of Gaussian Processes with Random Variance." Electron. J. Probab. 16 1254 - 1280, 2011. https://doi.org/10.1214/EJP.v16-904