On the asymptotics of supremum distribution for some iterated processes

In this paper, we study the asymptotic behavior of supremum distribution of some classes of iterated stochastic processes \(\{X(Y(t)) : t \in 0, \infty )\}\), where \(\{X(t) : t \in \mathbb {R} \}\) is a centered Gaussian process and \(\{Y(t): t \in 0, \infty )\}\) is an independent of {X(t)} stochastic process with a.s. continuous sample paths. In particular, the asymptotic behavior of \(\mathbb {P}(\sup _{s\in 0,T]} X(Y (s)) > u)\) as \(u \to \infty \), where T>0, as well as \(\lim _{u\to \infty } \mathbb {P}(\sup _{s\in 0,h(u)]} X(Y (s)) > u)\), for some suitably chosen function h(u) are analyzed. As an illustration, we study the asymptotic behavior of the supremum distribution of iterated fractional Brownian motion process.