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 _{sin [0,T]} X(Y (s)) > u)) as (u to infty ), where T>0, as well as (lim _{uto infty } mathbb {P}(sup _{sin [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.