New results on slowly varying functions in the Zygmund sense

Omey, Edward; Cadena, Meitner

doi:10.3792/pjaa.96.009

Cited by 3 publications

(6 citation statements)

References 11 publications

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“…in y, are presented. This limit generalizes the ones analyzed by Seneta [4] and Omey and Cadena [5], both of them being related to the monotony of functions in the Zygmund sense. Under this analysis, properties of θ(y) are described.…”

Section: Discussionsupporting

confidence: 83%

“…In [5], we found that relations of the form in Equation ( 1) hold with limit function θ(x) = 0. In that case, we have…”

Section: Third Formmentioning

confidence: 97%

“…As usual, we assume that g ∈ SN, r ∈ Γ 0 (g) and r(x) → ∞. From Theorem 3 in [5], we get the following representation:…”

Section: Third Formmentioning

confidence: 99%

“…More recently, Omey and Cadena's [5] functions extended the results of Seneta, and they considered functions for which the following relation holds:…”

Section: Introductionmentioning

confidence: 99%

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On Convergence Rates of Some Limits

Omey¹,

Cadena

2020

Mathematics

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In 2019 Seneta has provided a characterization of slowly varying functions L in the Zygmund sense by using the condition, for each y > 0 , x L ( x + y ) L ( x ) − 1 → 0 as x → ∞ . Very recently, we have extended this result by considering a wider class of functions U related to the following more general condition. For each y > 0 , r ( x ) U ( x + y g ( x ) ) U ( x ) − 1 → 0 as x → ∞ , for some functions r and g. In this paper, we examine this last result by considering a much more general convergence condition. A wider class related to this new condition is presented. Further, a representation theorem for this wider class is provided.

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Section: Discussionsupporting

confidence: 83%

“…In [5], we found that relations of the form in Equation ( 1) hold with limit function θ(x) = 0. In that case, we have…”

Section: Third Formmentioning

confidence: 97%

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On Convergence Rates of Some Limits

Omey¹,

Cadena

2020

Mathematics

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“…Nowadays, Kevei in [Kevei(2021)] presented an extension of such a conjecture and its proof. Coincidentally, we have been working on such a subject as a part of other researches related to results shown by [Seneta(2019)], see [Omey and Cadena(2020a)] and [Omey and Cadena(2020b)]. Unlike Kevei's proofs, ours are based on mainly the Williamson transform.…”

Section: Msc Code: 26a12 60e05mentioning

confidence: 97%

Analizing a Seneta's conjecture by using the Williamson transform

Omey¹,

Cadena²

2022

Preprint

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Truncated Moments for Heavy-Tailed and Related Distribution Classes

2023

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Suppose that ξ+ is the positive part of a random variable defined on the probability space (Ω,F,P) with the distribution function Fξ. When the moment Eξ+p of order p>0 is finite, then the truncated moment F¯ξ,p(x)=min1,Eξp1I{ξ>x}, defined for all x⩾0, is the survival function or, in other words, the distribution tail of the distribution function Fξ,p. In this paper, we examine which regularity properties transfer from the distribution function Fξ to the distribution function Fξ,p and which properties transfer from the function Fξ,p to the function Fξ. The construction of the distribution function Fξ,p describes the truncated moment transformation of the initial distribution function Fξ. Our results show that the subclasses of heavy-tailed distributions, such as regularly varying, dominatedly varying, consistently varying and long-tailed distribution classes, are closed under this truncated moment transformation. We also show that exponential-like-tailed and generalized long-tailed distribution classes, which contain both heavy- and light-tailed distributions, are also closed under the truncated moment transformation. On the other hand, we demonstrate that regularly varying and exponential-like-tailed distribution classes also admit inverse transformation closures, i.e., from the condition that Fξ,p belongs to one of these classes, it follows that Fξ also belongs to the corresponding class. In general, the obtained results complement the known closure properties of distribution regularity classes.

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New results on slowly varying functions in the Zygmund sense

Cited by 3 publications

References 11 publications

On Convergence Rates of Some Limits

On Convergence Rates of Some Limits

Analizing a Seneta's conjecture by using the Williamson transform

Truncated Moments for Heavy-Tailed and Related Distribution Classes

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