Second order tail behaviour of a subordinated probability distribution

Geluk, Jaap

doi:10.1016/0304-4149(92)90017-k

Cited by 18 publications

(15 citation statements)

References 8 publications

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“…We further mention Willekens (1986, 1987) who obtained second-order results. We also mention Geluk (1992Geluk ( , 1996 who provides second order results for subordinated probability distributions in the heavy tail case. Let N be a nonnegative random variable, independent of the X i 's.…”

Section: Proof (I) We First Setmentioning

confidence: 99%

Asymptotic expansions for infinite weighted convolutions of heavy tail distributions and applications

Barbe¹,

McCormick²

2009

Memoirs of the AMS

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Section: Proof (I) We First Setmentioning

confidence: 99%

Asymptotic expansions for infinite weighted convolutions of heavy tail distributions and applications

Barbe¹,

McCormick²

2009

Memoirs of the AMS

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“…See [16], [17] and [9] for more details aboutâ 2 (s). If f (s) is not continuous, but f is of bounded total variation, it follows from Grübel [20] that if there exists a monotonically decreasing function τ (s) with…”

Section: Asymptotic Approximations Of Compound Sumsmentioning

confidence: 99%

Higher-order expansions for compound distributions and ruin probabilities with subexponential claims

Albrecher

Hipp

Kortschak

2010

Scandinavian Actuarial Journal

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Let X i (i = 1, 2, . . .) be a sequence of subexponential positive independent and identically distributed random variables. In this paper we offer two alternative approaches to obtain higher-order expansions of the tail of n i=1 X i and subsequently for ruin probabilities in renewal risk models with claim sizes X i . In particular, these emphasize the importance of the term P( n i=1 X i > s, max(X 1 , . . . , X n ) ≤ s/2) for the accuracy of the resulting asymptotic expansion of P( n i=1 X i > s). Furthermore, we present a more rigorous approach to the often suggested technique of using approximations with shifted arguments. The cases of a Pareto-type, Weibull and Lognormal distribution for X i are discussed in more detail and numerical investigations of the increase in accuracy by including higher-order terms in the approximation of ruin probabilities for finite realistic ranges of s are given.

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“…In the case of independent random variables second order regular variation can be used to get higher order asymptotic approximation (c.f. Geluk 1992;Geluk et al 1997). Note that Eq.…”

Section: Assumption 21 There Exits Aĉmentioning

confidence: 99%

Second order tail asymptotics for the sum of dependent, tail-independent regularly varying risks

Kortschak

2011

Extremes

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In this paper we consider dependent random variables with common regularly varying marginal distribution. Under the assumption that these random variables are tail-independent, it is well known that the tail of the sum behaves like in the independence case. Under some conditions on the marginal distributions and the dependence structure (including Gaussian copula's and certain Archimedean copulas) we provide the second-order asymptotic behavior of the tail of the sum.

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Second order tail behaviour of a subordinated probability distribution

Cited by 18 publications

References 8 publications

Asymptotic expansions for infinite weighted convolutions of heavy tail distributions and applications

Asymptotic expansions for infinite weighted convolutions of heavy tail distributions and applications

Higher-order expansions for compound distributions and ruin probabilities with subexponential claims

Second order tail asymptotics for the sum of dependent, tail-independent regularly varying risks

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