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Point process of clusters for a stationary Gaussian random field on a lattice

Lu Yingyin, Guo Jinhui

Probability and Mathematical Statistics

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2023

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Vol. 43, Fasc. 2

247--262

EN

It is well established that the normalized exceedances resulting from a standard stationary Gaussian triangular array at high levels follow a Poisson process under the Berman condition. To model frequent cluster phenomena, we consider the asymptotic distribution of the point process of clusters for a Gaussian random field on a lattice. Our analysis demonstrates that the point process of clusters also converges to a Poisson process in distribution, provided that the correlations of the Gaussian random field meet certain conditions. Additionally, we provide a numerical example to illustrate our theoretical results.

2

Investigation of filtration capacity of surface wastewater filters by mathematical modelling

Marčiulaitienė Eglė, Meškauskaitė Laura, Pozniak Natalija, Sakalauskas Leonidas

Ecological Chemistry and Engineering. S

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2020

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Vol. 27, nr 2

241--255

EN

As the urbanisation level increases, due to intensification of car traffic and increased areas of impermeable surfaces, pollution of surface wastewater and a negative impact on water bodies are increasing. Due to the increasing pollution of surface water bodies, the eutrophication process is taking place intensively. One of the technologies of surface wastewater treatment allowing reduction in the amounts of suspended solids (SS), heavy metals and other pollutants is surface wastewater filters. Filters with different fillers have been designed for the treatment of principal surface wastewater pollutants: suspended solids, heavy metals (zinc, cadmium, copper, lead), BOD5, total carbon and nitrogen. The Kriging method was adapted to test the efficiency of filters filled with construction waste and wood waste-derived biochar using distance matrices. The method developed makes it possible to model the characteristics of filters in relation to different fillers, using experimental results. The mathematical model is suitable for other filtrate characteristics, not only the ratio of fillers, but also the length of the filter life, its durability calculations, which allows optimizing filter cleaning efficiency up to 96.93 %.

3

Extremes of multidimensional stationary Gaussian random fields

Soja-Kukieła N.

Probability and Mathematical Statistics

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2018

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Vol. 38, Fasc. 1

191--207

EN

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)).

4

Extremes of chi-square processes with trend

Liu P., Ji L.

Probability and Mathematical Statistics

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2016

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Vol. 36, Fasc. 1

1--20

EN

This paper studies the supremum of chi-square processes with trend over a threshold-dependent-time horizon. Under the assumptions that the chi-square process is generated from a centered self-similar Gaussian process and the trend function is modeled by a polynomial function, we obtain the exact tail asymptotics of the supremum of the chi-square proces with trend. These results are of interest in applications in engineering, insurance, queueing and statistics, etc. Some possible extensions of our results are also discussed.