Subexponentiality and infinite divisibility

Embrechts, Paul; Goldie, Charles M.; Veraverbeke, Noël

doi:10.1007/bf00535504

Cited by 316 publications

(222 citation statements)

References 13 publications

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“…Here we will directly use Theorem 4.2. We still have E M 0 < for all > 0, so we only need to check the regular variation of the function g in (9). In fact, we will prove that…”

Section: 3mentioning

confidence: 91%

“…A stochastic domination argument and the fact that, if the Lévy measure of an infinitely divisible random variable has a subexponential tail, then the distributional tail of the random variable is asymptotically equivalent to the tail of the Lévy measure (see Embrechts et al [9]), show that for large z P I * 0 > z ≤ Cz F Z z Therefore, as in the case of the infinite-source Poisson model, we conclude that (65) holds if (73) does. This completes the proof.…”

Section: A Renewal Poisson Cluster Processmentioning

confidence: 99%

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Scaling Limits for Cumulative Input Processes

Mikosch

Samorodnitsky

2007

Mathematics of OR

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We study different scaling behavior of very general telecommunications cumulative input processes. The activities of a telecommunication system are described by a marked-point process T n Z n n∈ , where T n is the arrival time of a packet brought to the system or the starting time of the activity of an individual source, and the mark Z n is the amount of work brought to the system at time T n . This model includes the popular ON/OFF process and the infinite-source Poisson model. In addition to the latter models, one can flexibly model dependence of the interarrival times T n − T n−1 , clustering behavior due to the arrival of an impulse generating a flow of activities, but also dependence between the arrival process T n and the marks Z n . Similarly to the ON/OFF and infinite-source Poisson model, we can derive a multitude of scaling limits for the input process of one source or for the superposition of an increasing number of such sources. The memory in the input process depends on a variety of factors, such as the tails of the interarrival times or the tails of the distribution of activities initiated at an arrival T n , or the number of activities starting at T n . It turns out that, as in standard results on the scaling behavior of cumulative input processes in telecommunications, fractional Brownian motion or infinite-variance Lévy stable motion can occur in the scaling limit. However, the fractional Brownian motion is a much more robust limit than the stable motion, and many other limits may occur as well.

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“…Here we will directly use Theorem 4.2. We still have E M 0 < for all > 0, so we only need to check the regular variation of the function g in (9). In fact, we will prove that…”

Section: 3mentioning

confidence: 91%

Section: A Renewal Poisson Cluster Processmentioning

confidence: 99%

Scaling Limits for Cumulative Input Processes

Mikosch

Samorodnitsky

2007

Mathematics of OR

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“…When α = 0, S := S (0) is the class of subexponential distributions. The subexponential or, more generally, the convolution equivalent distributions are used in modelling heavy-tailed data, such as occur in insurance applications; we refer the reader to [5], [6], [10], and the references therein for discussion and properties. Distributions in the class S (α) for α ≥ 0 are 'near to exponential' in the sense that their tails are only slightly modified exponential.…”

mentioning

confidence: 99%

Moment and MGF convergence of overshoots and undershoots for Lévy insurance risk processes

Park

Maller

2008

Adv. Appl. Probab.

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This paper is concerned with the finiteness and large-time behaviour of moments of the overshoot and undershoot of a high level, and of their moment generating functions (MGFs), for a Lévy process which drifts to −∞ almost surely. This provides information relevant to quantities associated with the ruin of an insurance risk process. Results of Klüppelberg, Kyprianou, and Maller (2004) and Doney and Kyprianou (2006) for asymptotic overshoot and undershoot distributions in the class of Lévy processes with convolution equivalent canonical measures are shown to have moment and MGF convergence extensions.

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“…If the boundedness of G/F in Theorem 1 is strengthened to the requirement lim^a, G(x)/F(x) = 0 then the condition Fe£C can be omitted (Embrechts et al (1979), Proposition 1). However this is not so in general, for in some circumstances the conclusion F*Ge£f implies Fe£C, as follows.…”

Section: Note That F(x)+g(x)~f(x)+g(x)-f(x)g(x)mentioning

confidence: 99%

“…The classes ^_ r Sf, 3) are closed under convolution roots (see Embrechts et al (1979) for the first two; the case of 3) is elementary). We conjecture that for each y > 0, Z£ y is closed under convolution roots: F ((I) eif y => ?…”

Section: Theorem 2 Let Fey Gesf and H = F*g Then Limsup^h L2 \X)mentioning

confidence: 99%