Product Convolution of Generalized Subexponential Distributions

Mikutavičius, Gustas; Šiaulys, Jonas

doi:10.3390/math11010248

Cited by 5 publications

(9 citation statements)

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“…On the other hand, it would be interesting to study the closure properties of randomly stopped structures that are not related to the sum or maximum, but related to the product of random variables, as was performed in paper [60], for instance. In the class of generalized subexponential distributions, this would be easier compared to other classes due to the results obtained in article [26].…”

Section: Discussionmentioning

confidence: 99%

“…For distributions of the nonnegative r.v.s, class OS was introduced by Klüppelberg [18], and later, for real-valued r.v.s, was studied by Yu and Wang [10], Shimura and Watanabe [9], Baltr ūnas et al [19], Watanabe and Yamamuro [20], Cheng and Wang [21], Lin and Wang [22], Hägele and Lehtomaa [23], Konstantinides et al [24],Yao and Taimre [25] and Mikutavičius and Šiaulys [26], among others.…”

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confidence: 99%

“…Moreover, in that paper, the closure under minimum is established, which means that F ξ 1 , F ξ 2 ∈ OS for independent r.v.s ξ 1 , ξ 2 imply F ξ 1 ∧ξ 2 ∈ OS. The authors of articles [24,26] consider when the distribution of the product of two independent random variables ξ, θ belongs to the class OS. For instance, in [26], it is proved that d.f.…”

mentioning

confidence: 99%

“…The authors of articles [24,26] consider when the distribution of the product of two independent random variables ξ, θ belongs to the class OS. For instance, in [26], it is proved that d.f. F ξθ is generalized subexponential if F ξ ∈ OS and θ is independent of ξ, nonnegative and not degenerated at zero.…”

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confidence: 99%

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Randomly Stopped Minimum, Maximum, Minimum of Sums and Maximum of Sums with Generalized Subexponential Distributions

Karasevičienė,

Šiaulys

2024

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In this paper, we find conditions under which distribution functions of randomly stopped minimum, maximum, minimum of sums and maximum of sums belong to the class of generalized subexponential distributions. The results presented in this article complement the closure properties of randomly stopped sums considered in the authors’ previous work. In this work, as in the previous one, the primary random variables are supposed to be independent and real-valued, but not necessarily identically distributed. The counting random variable describing the stopping moment of random structures is supposed to be nonnegative, integer-valued and not degenerate at zero. In addition, it is supposed that counting random variable and the sequence of the primary random variables are independent. At the end of the paper, it is demonstrated how randomly stopped structures can be applied to the construction of new generalized subexponential distributions.

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Section: Discussionmentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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confidence: 99%

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Randomly Stopped Minimum, Maximum, Minimum of Sums and Maximum of Sums with Generalized Subexponential Distributions

Karasevičienė,

Šiaulys

2024

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“…2. The closure property of class P D follows from the previous part 1., while the closure property of class OS is implied by [45].…”

Section: It Is Easy To See Thatmentioning

confidence: 99%

On the non-closure under convolution for strong subexponential distributions

Konstantinides

Leipus²,

Šiaulys³

2022

NAMC

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In this paper, we consider the convolution closure problem for the class of strong subexponential distributions, denoted as S*. First, we show that, if F, G ∈ L, then inclusions of F*G, FG, and pF + (1 – p)G for all (some) p ∈ (0; 1) into the class S* are equivalent. Then, using examples constructed by Klüppelberg and Villasenor [The full solution of the convolution closure problem for convolution-equivalent distributions, J. Math. Anal. Appl., 41:79–92, 1991], we show that S* is not closed under convolution.

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On the distribution-tail behaviour of the product of normal random variables

et al. 2023

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In this paper we consider the product $\Pi _{n}=\prod_{k=1}^{n}\xi _{k}$ Π n = ∏ k = 1 n ξ k of n independent normally distributed zero mean random variables $\xi _{1},\dots ,\xi _{n}$ ξ 1 , … , ξ n . We derive an asymptotic formula for the survival probability ${\mathbb{P}}(\Pi _{n}>x)$ P ( Π n > x ) , as $x\to \infty $ x → ∞ , with the first remaining term.

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Product Convolution of Generalized Subexponential Distributions

Cited by 5 publications

References 50 publications

Randomly Stopped Minimum, Maximum, Minimum of Sums and Maximum of Sums with Generalized Subexponential Distributions

Randomly Stopped Minimum, Maximum, Minimum of Sums and Maximum of Sums with Generalized Subexponential Distributions

On the non-closure under convolution for strong subexponential distributions

On the distribution-tail behaviour of the product of normal random variables

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