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 $\{Y(t):t\in [0,\mathrm{\infty })\}$ 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$T > 0$,aswellas$\lim_{u\to\infty} $\mathbb{P}$\left(\sup_{s \in [0, h(u)]} X(Y(s)) > u\right)$,forsomesuitablychosenfunction$h(u)$ are analyzed. As an illustration, we study the asymptotic behavior of the supremum distribution of iterated fractional Brownian motion process.