, Volume 20, Issue 2, pp 451–474 | Cite as

On the asymptotics of supremum distribution for some iterated processes

  • Marek Arendarczyk
Open Access


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


Exact asymptotics Supremum distribution Iterated process Iterated fractional brownian motion Gaussian process 

AMS 2000 Subject Classifications

Primary 60G15 60G18 Secondary 60G70 


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Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of WrocławWrocławPoland

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