Some properties and estimates for φ-sub-Gaussian stochastic processes

Vasylyk, Olga; Hopkalo, Olha; Kozachenko, Yuriy; Sakhno, Lyudmyla

doi:10.17721/1812-5409.2019/4.3

Cited by 3 publications

(5 citation statements)

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“…Example 4.5. One particular way to construct a ϕ-sub-Gaussian stochastic process was presented in Kozachenko and Koval'chuk (1998) (see also Vasylyk et al (2008)). Let {ξ k , k = 1, ∞} be a family of independent ϕ-sub-Gaussian random variables and ϕ be a such function that ϕ( √ x), x > 0, is convex.…”

Section: Applicationsmentioning

confidence: 99%

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Bounds for the Tail Distributions of Suprema of Sub-Gaussian Type Random Fields

Hopkalo,

Sakhno,

Vasylyk

2023

AJS

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The paper presents bounds for the distributions of suprema for particular classes of ϕ-sub-Gaussian random fields. Results stated depend on representations of bounds for increments of the fields in different metrics. Several examples of applications are provided to illustrate the results, in particular, to random fields related to stochastic partial differential equations and partial differential equations with random initial conditions.

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Section: Applicationsmentioning

confidence: 99%

“…The generalization of this notion to the classes of ϕ-sub-Gaussian random variables is introduced as follows (see, Buldygin and Kozachenko (2000) (Ch.2), Giuliano, Kozachenko, and Nikitina (2003), Kozachenko and Ostrovskij (1986), Vasylyk, Kozachenko, and Yamnenko (2008)).…”

Section: Introductionmentioning

confidence: 99%

Bounds for the Tail Distributions of Suprema of Sub-Gaussian Type Random Fields

Hopkalo,

Sakhno,

Vasylyk

2023

AJS

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“…Sub ϕ (Ω) spaces (spaces of ϕ-sub-Gaussian random variables) are spaces of centered random variables with certain exponential moments. For a more in-depth understanding of these spaces, refer to the book of Vasylyk, Kozachenko, and Yamnenko (2008). Additionally, Kozachenko and Vasylyk (2001) introduced more general classes V (ϕ, ψ) of random processes.…”

Section: Introductionmentioning

confidence: 99%

“…The paper is organized as follows. Section 2 is devoted to the general theory of random variables and processes from Orlicz spaces of exponential type and based on the works Buldygin and Kozachenko (2000); Kozachenko and Vasylyk (2001); Vasylyk et al (2008); Vasylyk and Yamnenko (2007); Yamnenko, Kozachenko, and Bushmitch (2014). Section 3 contains a generalization of results from the papers Kozachenko and Yamnenko (2014) and the last section contains applications to generalized Wiener and FBM processes from classes V (ϕ, ψ).…”

Section: Introductionmentioning

confidence: 99%

Supremum Distribution of Weighted Sum of Random Processes from Orlicz Spaces of Exponential Type with Continuous Deviation

Tykhonenko,

Yamnenko

2023

AJS

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The paper studies distribution of sum of random processes from Orlicz spaces of exponential type weighted by continuous functions, in particular, processes from spaces Subϕ (Ω), SSubϕ (Ω) and class V (ϕ, ψ) are considered. Such spaces and classes of random variables and corresponding stochastic processes provide generalizations of Gaussian and sub-Gaussian random variables and processes and are important for various applications, for example, in queuing theory and financial mathematics. We derive the estimates for the distribution of supremum of weighted sum of such processes deviated by a continuous monotone function using the entropy method. As examples, weighted sum of Wiener and weighted sum of fractional Brownian motion processes with different Hurst indices from classes V (ϕ, ψ) are considered. Corresponding estimates of the probability of exceeding by trajectories of such weighted sums a positive level determined by a linear function are obtained. In the insurance risk theory, such aproblem arises during estimating a ruin probability of the corresponding risk process with a constant premium income, and in the communications theory, it appears for the buffer overflow probability for a single server with a constant service rate.

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“…The generalization of this notion to the classes of ϕ-sub-Gaussian random variables is introduced as follows (see, [4,Ch.2], [7], [16], [20]).…”

Section: Introductionmentioning

confidence: 99%

Properties of solutions to linear KdV equations with φ-sub-Gaussian initial conditions

Hopkalo

Sakhno

Vasylyk

2022

BKNUPhM

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In this paper, there are studied sample paths properties of stochastic processes representing solutions (in L_2(Ω) sense) to the linear Korteweg–de Vries equation (called also the Airy equation) with random initial conditions given by φ-sub-Gaussian stationary processes. The main results are the bounds for the distributions of the suprema for such stochastic processes considered over bounded domains. Also, there are presented some examples to illustrate the results of the study.

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Some properties and estimates for φ-sub-Gaussian stochastic processes

Cited by 3 publications

References 1 publication

Bounds for the Tail Distributions of Suprema of Sub-Gaussian Type Random Fields

Bounds for the Tail Distributions of Suprema of Sub-Gaussian Type Random Fields

Supremum Distribution of Weighted Sum of Random Processes from Orlicz Spaces of Exponential Type with Continuous Deviation

Properties of solutions to linear KdV equations with φ-sub-Gaussian initial conditions

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