Extremes of Gaussian processes with a smooth random variance
Three main theorems are established in this paper. Let \((X(t))_{t\in\mathbb R}\) be a random process, whose conditional distribution, given the random element \(Y\), is Gaussian. The first main result provides the asymptotics of \(\text{P}(\sup_{t\in[0,T]}X(t)>u)\) as \(u\to\infty\). Let \((\xi(t))_{t\in\mathbb R}\) be a Gaussian process and set \(\eta(t):=\lambda-\zeta t^2/2\), \(t\in\mathbb R\), where \(\lambda\), \(\zeta\) are non-negative random variables being independent of \(\xi(\cdot)\). The second main result establishes the asymptotic behavior of \(\text{P}(\sup_{t\in[-T,T]}\xi(t)(\lambda-\xi t^2/2)>u)\) as \(u\to\infty\). The third main result provides the asymptotics of \(\text{P}(\sup_{t\in[0,T]}\xi(t)\eta(t)>u)\) as \(u\to\infty\) by extending the particular parabola process from Theorem 2 to smooth processes \(\eta(\cdot)\). Each result is, of course, established under appropriate conditions.
scientific article; zbMATH DE number 5956357