Asymptotic expansions for infinite weighted convolutions of rapidly varying subexponential distributions

Barbe, Philippe; McCormick, William P.

doi:10.1007/s00440-007-0082-1

Cited by 8 publications

(10 citation statements)

References 38 publications

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“…Following another approach, Barbe & McCormick [7,10] (see also [8,9]) recently used the concept of smoothly varying functions to derive higher-order expansions: …”

Section: Asymptotic Approximations Of Compound Sumsmentioning

confidence: 99%

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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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“…Following another approach, Barbe & McCormick [7,10] (see also [8,9]) recently used the concept of smoothly varying functions to derive higher-order expansions: …”

Section: Asymptotic Approximations Of Compound Sumsmentioning

confidence: 99%

“…The proofs for F (s) ∈ R are more involved (for details see [10]). In [7] Laplace characters are defined by…”

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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“…f (x)ES 2 n−1 + · · · , cf. [24], [11], [10] and [9]. Technically, the Taylor expansion is only useful for moderate S n−1 , and large values have to be shown to be negligible by a separate argument; this also is the case in the present paper.…”

Section: Introductionmentioning

confidence: 73%

Error Rates and Improved Algorithms for Rare Event Simulation with Heavy Weibull Tails

Asmussen

Kortschak

2013

Methodol Comput Appl Probab

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Let Y 1 , . . . , Y n be i.i.d. subexponential and S n = Y 1 + · · · + Y n . Asmussen and Kroese (2006) suggested a simulation estimator for evaluating P(S n > x), combining an exchangeability argument with conditional Monte Carlo. The estimator was later shown by Hartinger & Kortschak (2009) to have vanishing relative error. For the Weibull and related cases, we calculate the exact error rate and suggest improved estimators. These improvements can be seen as control variate estimators, but are rather motivated by second order subexponential theory which is also at the core of the technical proofs.

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“…Recent studies on distributions with rapidly-varying tails can be found in Tang and Tsitsiashvili (2004) and Barbe and McCormick (2008). Clearly, all distributions in the class L(γ) for γ > 0 belong to the class R −∞ .…”

Section: Introduction and Main Resultsmentioning

confidence: 97%

From light tails to heavy tails through multiplier

Tang

2008

Extremes

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Let X and Y be two independent nonnegative random variables, of which X has a distribution belonging to the class L(γ) or S(γ) for some γ ≥ 0 and Y is unbounded. We study how their product XY inherits the tail behavior of X. Under some mild technical assumptions we prove that the distribution of XY belongs to the class L(0) or S(0) accordingly. Hence, the multiplier Y builds a bridge between light tails and heavy tails.

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Asymptotic expansions for infinite weighted convolutions of rapidly varying subexponential distributions

Cited by 8 publications

References 38 publications

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

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

Error Rates and Improved Algorithms for Rare Event Simulation with Heavy Weibull Tails

From light tails to heavy tails through multiplier

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