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Extremes of order statistics of stationary Gaussian processes

Zhao C.

Probability and Mathematical Statistics

2018

Vol. 38, Fasc. 1

61--75

Let {Xi(t), t ≥ 0}, 1 ≤ i ≤ n, be mutually independent and identically distributed centered stationary Gaussian processes. Under some mild assumptions on the covariance function, we derive an asymptotic expansion of P [formula] ]X(r) (t) ≤ u) as u → ∞, where mr(u) = (P([formula] X(r) (t) > u))−1 (1 + o(1)), and {X(r) (t), t ≥ 0} is the rth order statistic process of {Xi(t), t ≥ 0}, 1 ≤ i, r ≤ n. As an application of the derived result, we analyze the asymptotics of supremum of the order statistic process of stationary Gaussian processes over random intervals.

Extremes of multidimensional stationary Gaussian random fields

Soja-Kukieła N.

Probability and Mathematical Statistics

2018

Vol. 38, Fasc. 1

191--207

Let {X(t) : t = (t1, t2,…, td) ϵ [0, ∞)d} be a centered stationary Gaussian field with almost surely continuous sample paths, unit variance and correlation function r satisfying r(t) < 1 for every t ≠ 0 and r(t) = 1 – Σdi=1 |ti|αi + o (Σdi=1 |ti|αi), as t → 0, with some α1, α2,…, αd ϵ (0, 2]. The main result of this contribution is the description of the asymptotic behaviour of P (sup{X(t) : t ϵ Jxm} ≤ u), as u → ∞, for some Jordan-measurable sets Jxm of volume proportional to P (sup{X(t) : t ϵ [0, 1]d} > u)−1 (1 + o(1)).

On Usability of Ordering and Equivalence Relations to Define and to Ground Some Mathematical Notions in Secondary Schools

Bryll G., Sochacki R.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

2003

Vol. 9

131--136

Many important mathematical notions can be introduced by means of the notions of supremum, infimum and equivalence class. Our considerations refer to mainly pupils of mathematical sections.

Distributions of suprema of Lévy processes via the Heavy Traffic Invariance Principle

Szczotka W., Woyczyński W. A.

Probability and Mathematical Statistics

2003

Vol. 23, Fasc. 2

251--272

We study the relationship between the distribution of the supremum functional MX = sup0 ≤ t < ∞ (X(t) − βt) for a process X with stationary, but not necessarily independent increments, and the limiting distribution of an appropriately normalized stationary waiting time for G/G/l queues in heavy traffic. As a by-product we obtain explicit expressions for the distribution of MX in several special cases of Lévy processes.