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 ∈ [0, ∞)}, where {X(t) : t ∈ R} is a centered Gaussian process and {Y (t) : t ∈ [0, ∞)} is an independent of {X(t)} stochastic process with a.s. continuous sample paths. In particular, the asymptotic behavior of P(sup s∈[0,T ] X(Y (s)) > u) as u → ∞, where T > 0, as well as lim u→∞ P(sup s∈ [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.

Extremes of randomly scaled Gumbel risks

“…As an illustration of Theorems 1.1 and 2.1, we shall analyze: a) limiting behaviour of the maximum of randomly scaled Gaussian processes; b) exact asymptotic tail behaviour of the supremum of Gaussian processes with stationary increments over a random interval with length which has Weibullian tail behaviour; c) exact asymptotic tail behaviour of sup t∈[0,T ] X(Y (t)) with X a centered Gaussian processes with stationary increments being independent of Y , extending the recent findings of [5].…”

“…Let Y (t), t ≥ 0 be a random process independent of X with continuous sample paths. In view of [5][Theorem 2.1], if both…”

“…In view of our Theorem 2.1, statement (b), this is not always the case. In the following we present an extension of Theorem 2.1 in [5], which in particular is applicable for the analysis of sup t∈[0,T ] X(SY (t)) when S has a log-Weibullian asymptotic tail behaviour.…”

“…The study of products of rvs is of interest for numerous applications, see e.g., [1][2][3][4][5][6][7][8][9][10][11][12][13]. We mention below three with Y 1 > 0 being independent of Y 2 .…”

“…If X is a more general Gaussian process, for instance X being centred with stationary increments the direct relation in (1.3) does not hold. In view of [5][Theorem…”