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.