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 \(\mathbb {P}(\sup _{s\in [0,T]} X(Y (s)) > u)\) 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.
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Arendarczyk, M. On the asymptotics of supremum distribution for some iterated processes. Extremes 20, 451–474 (2017). https://doi.org/10.1007/s10687-016-0272-2
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DOI: https://doi.org/10.1007/s10687-016-0272-2
Keywords
- Exact asymptotics
- Supremum distribution
- Iterated process
- Iterated fractional brownian motion
- Gaussian process