Tail asymptotics of light-tailed Weibull-like sums

Asmussen, Søren; Hashorva, Enkelejd; Laub, Patrick J.; Taimre, Thomas

doi:10.19195/0208-4147.37.2.3

Cited by 15 publications

(10 citation statements)

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“…The asymptotic form for the heavy-tailed Weibull sum is covered by (5.6) and (5.7) as the summands are subexponential. The difficulty in finding the asymptotics for the light-tailed case led the authors to investigate it in detail, leading to the paper [17] which uses results originally from [25].…”

Section: The Radial Approximationmentioning

confidence: 99%

“…The exposition in [17] details this and more general asymptotics (i.e. the independent but non-identically distributed case, and when the variables are not exactly Weibull but are 'Weibull-like').…”

Section: The Radial Approximationmentioning

confidence: 99%

“…One (light-tailed) case where we can determine an asymptotic angular distribution is for light-tailed Weibull sums. The angular asymptotic can be extracted from the following result in [17].…”

Section: Algorithmmentioning

confidence: 99%

“…The specific gamma distribution is moment-matched with the asymptotic normal approximation for the exponentially tilted Weibull distribution, cf. Section 6 of [17].…”

Section: Light-tailed Weibull Summandsmentioning

confidence: 99%

“…Theorem 6. 17. Let (X i , X j ) be a pair from a type I elliptical random vector X ∼ E(µ, Σ, F ) and consider γ x F .…”

Section: A2 Asymptotic Properties Of Type I Elliptical Distributionsmentioning

confidence: 99%

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Computational methods for sums of random variables

Laub¹

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A typical problem in the field of rare-event estimation is to find the probability P(S > γ) where S := X 1 + · · · + X d for a fixed d ∈ N + and where the γ ∈ R is large or increasing. In applications we often wish to understand the behaviour of a combination of random factors. Hence the random variable S is ubiquitous in real-world modeling problems. It can model, for example, aggregate risk or portfolio value for holding d risky assets [124,152], the aggregate losses for d insurance policy claims [14,107], and the combined signal interference from d wireless transmission sources [73]. Probabilities of this form are used to understand how a system would behave under extreme scenarios such as a market crash, a power surge, or a natural disaster. One is typically interested in, not just the quantity P(S > γ), but the behaviour of the summands when the extreme event {S > γ} occurs.

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Section: The Radial Approximationmentioning

confidence: 99%

Section: The Radial Approximationmentioning

confidence: 99%

Section: Algorithmmentioning

confidence: 99%

“…The specific gamma distribution is moment-matched with the asymptotic normal approximation for the exponentially tilted Weibull distribution, cf. Section 6 of [17].…”

Section: Light-tailed Weibull Summandsmentioning

confidence: 99%

“…Theorem 6. 17. Let (X i , X j ) be a pair from a type I elliptical random vector X ∼ E(µ, Σ, F ) and consider γ x F .…”

Section: A2 Asymptotic Properties Of Type I Elliptical Distributionsmentioning

confidence: 99%

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Computational methods for sums of random variables

Laub¹

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Chapter III: Sums and Aggregate Claims

Asmussen

Steffensen

2020

Probability Theory and Stochastic Modelling

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Extremal dependence of random scale constructions

2019

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A bivariate random vector can exhibit either asymptotic independence or dependence between the largest values of its components. When used as a statistical model for risk assessment in fields such as finance, insurance or meteorology, it is crucial to understand which of the two asymptotic regimes occurs. Motivated by their ubiquity and flexibility, we consider the extremal dependence properties of vectors with a random scale construction (X1, X2) = R(W1, W2), with non-degenerate R > 0 independent of (W1, W2). Focusing on the presence and strength of asymptotic tail dependence, as expressed through commonly-used summary parameters, broad factors that affect the results are: the heaviness of the tails of R and (W1, W2), the shape of the support of (W1, W2), and dependence between (W1, W2). When R is distinctly lighter tailed than (W1, W2), the extremal dependence of (X1, X2) is typically the same as that of (W1, W2), whereas similar or heavier tails for R compared to (W1, W2) typically result in increased extremal dependence. Similar tail heavinesses represent the most interesting and technical cases, and we find both asymptotic independence and dependence of (X1, X2) possible in such cases when (W1, W2) exhibit asymptotic independence. The bivariate case often directly extends to higher-dimensional vectors and spatial processes, where the dependence is mainly analyzed in terms of summaries of bivariate sub-vectors. The results unify and extend many existing examples, and we use them to propose new models that encompass both dependence classes.

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Tail asymptotics of light-tailed Weibull-like sums

Cited by 15 publications

References 1 publication

Computational methods for sums of random variables

Computational methods for sums of random variables

Chapter III: Sums and Aggregate Claims

Extremal dependence of random scale constructions

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