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1

A limit theorem for the last exit time over a moving nonlinear boundary for a Gaussian process

Karagodin Nikita

Probability and Mathematical Statistics

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2022

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Vol. 42, Fasc. 2

195--217

EN

We prove the convergence of the distribution of the scaled last exit time over a slowly moving nonlinear boundary for a class of Gaussian stationary processes. The limit is a double exponential (Gumbel) distribution.

2

Occupation time problem for multifractional Brownian motion

Ouahra Mohamed Ait, Guerbaz Raby, Ouahhabi Hanae, Sghir Aissa

Probability and Mathematical Statistics

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2019

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Vol. 39, Fasc. 1

99--113

EN

In this paper, by using a Fourier analytic approach, we investigate sample path properties of the fractional derivatives of multifractional Brownian motion local times. We also show that those additive functionals satisfy a property of local asymptotic self-similarity. As a consequence, we derive some local limit theorems for the occupation time of multifractional Brownian motion in the space of continuous functions.

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

Diffusion limits for the queue length of jobs in multi-server open queueing networks

Greičius E., Minkevičius S.

Operations Research and Decisions

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2017

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Vol. 27, No. 4

71--84

EN

A mathematical model of a multiserver open queueing network in heavy traffic is developed. This model is that of a multiserver computer system network in heavy traffic. A limit theorem for the length of the queue has been presented.

5

Unconditional convergence in lattice groups with respect to ideals

Boccuto A., Dimitriou X., Papanastassiou N.

Commentationes Mathematicae

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2010

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Vol. 50, [Z] 2

161-174

EN

We deal with unconditional convergence of series and some special classes of subsets of N.

6

On the local time of multiparameter symmmetric stable processes. Regularity and limit theorem in Besov spaces

Boufoussi B., Dozzi M.

Probability and Mathematical Statistics

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2003

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Vol. 23, Fasc. 2

369--387

EN

Let X = (Xz, z ϵ TN = [0, l]N) be a symmetric α-stable process, 1 < α ≤ 2. Based on a Kolmogorov type continuity theorem we show Hölder conditions in Lp-norms for the local time of X with respect to the space and time variables, by distinguishing the cases where the time variables do or do not meet the axes. Weak convergence of the occupation integral is proved.

7

A central limit theorem for strictly stationary sequences in terms of slow variation in the limit

Szewczak Z. S.

Probability and Mathematical Statistics

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1998

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Vol. 18, Fasc. 2

359--368

EN

For strictly stationary random sequences satisfying the “minimal” dependence condition, necessary and sufficient conditions for the weak convergence to the normal law in terms of slow variation in the limit are found.