On the asymptotics of supremum distribution for some iterated processes

Abstract: 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 $\mathbb{P}\left(\sup_{s \in [0,T]} X(Y(s)) > u\right)$ as $u \to \infty$, where $T > 0$, as well as $\lim_{u\to\infty} \mathbb{P}\left(\sup_{s \in [0, h(u)]} X(Y(s)) > u\right)$, 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.