Pseudo-Regularly Varying Functions and Generalized Renewal Processes

Buldygin, V. V.; Indlekofer, Karl-Heinz; Klesov, Oleg; Steinebach, Josef

doi:10.1007/978-3-319-99537-3

Cited by 24 publications

(13 citation statements)

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“…Indeed, if u(x) := x β F (x) is a logarithmically periodic function for large x, meaning that for some p > 1 we have u(x) = u(px), then x 0 u(y)y −1 dy is slowly varying. See Lemma 2.3 by Kevei [6], or in a more general setting Proposition 6.7 by Buldygin et al [2]. The simplest such example for β = 1 is the classical St. Petersburg distribution, with distribution function On the other hand, if ρ = β, then h β ∈ RV β implies that F (x) is slowly varying and h β (x) ∼ F (x)x β .…”

Section: )mentioning

confidence: 99%

On a conjecture of Seneta on regular variation of truncated moments

Kevei

2021

Publ Inst Math (Belgr)

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We prove that h?(x) = ??x0 y??1F?(y)dy is regularly varying with index ? [0, ?) if and only if V?(x) = ?[0,x] y?dF(y) is regularly varying with the same index, where ? > 0, F(x) is a distribution function of a nonnegative random variable, and F?(x) = 1?F(x). This contains at ? = 0, ?= 1 a result of Rogozin [8] on relative stability, and at ? = 0, ? = 2 a new, equivalent characterization of the domain of attraction of the normal law. For ? = 0 and ? > 0 our result implies a recent conjecture by Seneta [9].

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Section: )mentioning

confidence: 99%

On a conjecture of Seneta on regular variation of truncated moments

Kevei

2021

Publ Inst Math (Belgr)

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“…Remind that a positive and measurable function is called a regularly varying with index ∈ R if for any > 0 lim →∞ ( ) ( ) = (see Karamata (1930); Seneta (1976)). Various classes of regularly varying functions and their extensions developed in a series of works by (V. V. Buldygin, Indlekofer, Klesov, & Steinebach, 2012 3 Asymptotic Behavior of Surplus Processes with Finite Sum of a Random Component…”

Section: Asymptotic Behavior Of Surplus Processesmentioning

confidence: 99%

Асимптотичні Властивості Процесів, Які Виникають У Задачах Оптимального Інвестування

Klesov

Tymoshenko

Pavlenkov

2019

MMTU

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“…У монографії [4] розглядається клас псевдорегулярних функцій та за допомогою властивостей таких функцій досліджуються узагальнені процеси відновлення. Зауважимо, що клас псевдорегулярних функцій тісно пов'язаний із класом правильно змінних функцій.…”

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The Regularization of Unitary Matrix Functions

Pavlenkov¹

2016

НВ НТУУ КПІ

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Background. The limit behavior at infinity of unitary matrix functions of real argument and arbitrary finite dimension is considered. The question of regular variation in Karamata sense of these functions is studied. Objective. The main aim of this work is to find the conditions under which not regularly varying unitary matrix function of real argument can be regularized by variable substitution. Methods. It is shown in the paper that substitution log t t  converts two-dimensional unitary matrix function into a power function with known matrix degree. This property is the base of all main result's proofs. Results. The conditions under which the unitary matrix function of arbitrary finite dimension can be regularized are obtained. Conclusions. A significant difference between the matrix functions of dimension lower than 4 and the matrix functions of higher dimension is established. The constructed example of unitary matrix function of 4-dimension that can't be regularized by substitution log t t  shows this difference.

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Pseudo-Regularly Varying Functions and Generalized Renewal Processes

Cited by 24 publications

References 0 publications

On a conjecture of Seneta on regular variation of truncated moments

On a conjecture of Seneta on regular variation of truncated moments

Асимптотичні Властивості Процесів, Які Виникають У Задачах Оптимального Інвестування

The Regularization of Unitary Matrix Functions

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