Infinitely divisible sequences

Hawkes, J. G.; Jenkins, J.D.

doi:10.1080/03461238.1978.10419477

Cited by 13 publications

(9 citation statements)

References 14 publications

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“…Since / ) ( A^ -k) -r n _ k q k and r n -> (EL + )"' (apply Theorem 4.2 of Hawkes et al (1978)) we have…”

Section: >N) and Q H = P(n~ = N) = P(l + > N)mentioning

confidence: 99%

“…In this paper our main concern is with the asymptotic behaviour of p n /a n , as opposed to that of (2 k>n Pk)/(^k»n <**)• Special cases have already been considered by Hawkes et al (1978) (in particular in Theorem 3.1). E. M. Wright (1967) works with the system of equations (3) but with the side conditions p o = 1, X = 1 and 2 a, = oo, he proved that the following statements are equivalent:…”

Section: (2)mentioning

confidence: 99%

“…= oo we cannot make use of the Chover, Ney and Wainger results. In this situation the results of Hawkes et al (1978) can be used to give information about the behaviour of p n .…”

mentioning

confidence: 99%

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A limit theorem for the tails of discrete infinitely divisible laws with applications to fluctuation theory

Embrechts¹,

Hawkes

1982

J Aust Math Soc A

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Suppose that (/>") is an infinitely divisible distribution on the non-negative integers having Levy measure (v n ). In this paper we derive a necessary and sufficient condition for the existence of the limit lim ; , -ooPn/"n-We also derive some other results on the asymptotic behaviour of the sequence (p n ) and apply some of our results to the theory of fluctuations of random walks. We obtain a necessary and sufficient condition for the first positive ladder epoch to belong to the domain of attraction of a spectrally positive stable law with index a, a G (1,2).1980 Mathematics subject classification (Amer. Math. Soc): 670 E 07, 60 J 15.

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“…Since / ) ( A^ -k) -r n _ k q k and r n -> (EL + )"' (apply Theorem 4.2 of Hawkes et al (1978)) we have…”

Section: >N) and Q H = P(n~ = N) = P(l + > N)mentioning

confidence: 99%

Section: (2)mentioning

confidence: 99%

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A limit theorem for the tails of discrete infinitely divisible laws with applications to fluctuation theory

Embrechts¹,

Hawkes

1982

J Aust Math Soc A

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“…The theory of infinitely divisible distributions began with de Finetti [4]. See, e.g., Wright [30], Hawkes and Jenkins [9], van Harn [28], Embrechts and Hawkes [5] and Steutel and van Harn [26] for background and results in the discrete case which are pertinent to the current article. Often, when two sequences are related in this way, asymptotic information can be transferred between them.…”

Section: Introductionmentioning

confidence: 99%

“…Often, when two sequences are related in this way, asymptotic information can be transferred between them. The purpose of this note is to point out that the asymptotics of S n can be obtained from those of N n , via the limit theory developed in [5,9]. See Section 3 for the proof of Theorem 1.…”

Section: Introductionmentioning

confidence: 99%

The asymptotic number of score sequences

Kolesnik¹

2022

Preprint

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A tournament on a graph is an orientation of its edges. The score sequence lists the in-degrees in non-decreasing order. Works by Winston and Kleitman (1983) and Kim and Pittel (2000) showed that the number S n of score sequences on the complete graph K n satisfies S n = Θ(4 n /n 5/2 ). In this work, by combining recent combinatorial developments related to score sequences with the limit theory for discrete infinitely divisible distributions, we observe that n 5/2 S n /4 n → 0.392 . . ., as conjectured by Takács (1986).

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The occurrence and frequency of 2n pollen in three diploid solanums from Northwest Argentina

Camadro

Peloquin

1980

Theoret. Appl. Genetics

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A group of wild, tuber-bearing species from Northwest Argentina, belonging to the series Tuberosa, Solarium spegazzini Bitt. (spg, 2n=2x=24), S. gourlayi Hawkes (grl, 2n=2x=24 and 2n=4x=48) and S. oplocense Hawkes (opl, 2n=6x=72), and Cuneolata, S. infundibuliforme Phil (ifd, 2n=2x=24), is being used to investigate the mode of origin of polyploids in the genus Solanum. 2n gametes have been detected in the diploid species ifd and spg and in a diploid race of grl, using cytological and breeding approaches. Twenty-two introductions of spg, 8 of grl and 26 of ifd have been tested for 2n pollen; 59%, 63% and 54% of them, respectively, had at least one 2n pollen producing plant. These introductions comprised 238, 76 and 235 plant respectively, of which 20, 16, and 32 plant produced 5% or more 2n pollen. The mechanism of 2n pollen formation was determined in several plant of 2x spg, 2x grl and 2x ifd. All of them were found to form diplandroids via parallel spindles. This mechanism, which gives meiotic products genetically equivalent to first division restitution gametes, is under control of the Mendelian recessive ps. The results suggest that the allele ps is widely distributed in natural populations of the three diploids, and that its frequency is very high. These species are seen as valuable material for population genetic studies, and for the eventual incorporation into a breeding scheme involving sexual polyploidization via 2n gametes.

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Infinitely divisible sequences

Cited by 13 publications

References 14 publications

A limit theorem for the tails of discrete infinitely divisible laws with applications to fluctuation theory

A limit theorem for the tails of discrete infinitely divisible laws with applications to fluctuation theory

The asymptotic number of score sequences

The occurrence and frequency of 2n pollen in three diploid solanums from Northwest Argentina

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