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 , where is a centered Gaussian process and is an independent of {X(t)} stochastic process with a.s. continuous sample paths. In particular, the asymptotic behavior of as , where T > 0, as well as , 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.