Renewal theory for extremal Markov sequences of Kendall type

Jasiulis-Gołdyn, B. H.; Misiewicz, Jolanta K.; Naskręt, K.; Omey, Edward

doi:10.1016/j.spa.2019.09.013

Cited by 11 publications

(22 citation statements)

References 21 publications

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“…Moreover, in[9, Theorem 3] it is also shown that h β is in fact slowly varying in the Zygmund sense, which is, by a result of Bojanić and Karamata [1, Theorem 1.5.5], equivalent to h β being normalized slowly varying.Remark 1.3. The implications (1.5) ⇒ (1.4), (1.6), (1.7), (1.8) are contained in Lemma 5 in Jasiulis-Gołdyn et al[5].Remark 1.4. For ρ ∈ (0, β) the functions x β F (x), h β (x), and V β (x) are all asymptotically equivalent up to a strictly positive finite constant factor.…”

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confidence: 87%

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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confidence: 87%

On a conjecture of Seneta on regular variation of truncated moments

Kevei

2021

Publ Inst Math (Belgr)

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“…This definition was extended to symmetrical measures on R by Jasiulis-Gołdyn in [12]. Generalized convolutions were explored with the use of regular variation ( [2,3,13]) and were used to construct Lévy processes and stochastic integrals ( [4], [28]). In the theory of generalized convolutions, we create new mathematical objects that have potential in applications.…”

Section: Introductionmentioning

confidence: 99%

“…generalized extreme value distribution (Frechét, Gumbel, Weibull) is commonly used for modeling rainfall, floods, drought, cyclones, extreme air pollutants, etc. Random walks with respect to generalized convolutions form a class of extremal Markov chains (see [1,4,13]). Studying them in the appropriate algebras will be a meaningful contribution to extreme value theory ( [5]).…”

Section: Introductionmentioning

confidence: 99%

“…Studying them in the appropriate algebras will be a meaningful contribution to extreme value theory ( [5]). Kendall random walks ( [11,13]), which are the main objects of investigations in this paper, are related to maximal processes (and thus maximum convolution), Pareto processes ( [8,32]) or pRARMAX processes ( [7]). One of the differences lies in the fact that in this case values of the process are randomly multiplied by a heavy-tailed Pareto random variables, which results in even more extreme behavior than in the case of the classical maximum process.…”

Section: Introductionmentioning

confidence: 99%

“…where measure ν ∈ P s is called the step distribution. Construction and some basic properties of this particular process are described in [4,11,13,14].…”

Section: Introductionmentioning

confidence: 99%

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Fluctuations of extremal Markov chains driven by the Kendall convolution

Jasiulis-Gołdyn¹,

Omey²,

Staniak³

2019

Preprint

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The paper deals with fluctuations of Kendall random walks, which are extremal Markov chains. We give the joint distribution of the first ascending ladder epoch and height over any level a ≥ 0 and distribution of maximum and minimum for these extremal Markovian sequences. We show that distribution of the first crossing time of level a ≥ 0 is a mixture of geometric and negative binomial distributions. The Williamson transform is the main tool for considered problems connected with the Kendall convolution.

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How Exceptional is the Extremal Kendall and Kendall-Type Convolution

Jasiulis-Gołdyn,

Misiewicz,

Omey

et al. 2023

Results Math

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This paper deals with the generalized convolutions connected with the Williamson transform and the maximum operation. We focus on such convolutions which can define transition probabilities of renewal processes. They should be monotonic since the described time or destruction does not go back, it should admit existence of a distribution with a lack of memory property because the analog of the Poisson process shall exist. Another valuable property is the simplicity of calculating and inverting the corresponding generalized characteristic function (in particular Williamson transform) so that the technique of generalized characteristic function can be used in description of our processes. The convex linear combination property (the generalized convolution of two point measures is the convex combination of several fixed measures), or representability (which means that the generalized convolution can be easily written in the language of independent random variables)—they also facilitate the modeling of real processes in that language. We describe examples of generalized convolutions having the required properties ranging from the maximum convolution and its simplest generalization—the Kendall convolution (associated with the Williamson transform), up to the most complicated here—Kingman convolution. It is novel approach to apply in the extreme value theory. Stochastic representation of the Kucharczak-Urbanik in the order statistics terms is proved, which open new paths to investigate Archimedean copulas. This paper open the door to solve an old open problem of the relationship between copulas and generalized convolutions mentioned by B. Schweizer and A. Sklar in 1983. This indicates the path of further research towards extremes and dependency modelling.

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Renewal theory for extremal Markov sequences of Kendall type

Cited by 11 publications

References 21 publications

On a conjecture of Seneta on regular variation of truncated moments

On a conjecture of Seneta on regular variation of truncated moments

Fluctuations of extremal Markov chains driven by the Kendall convolution

How Exceptional is the Extremal Kendall and Kendall-Type Convolution

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