Subexponentiality of the product of independent random variables

Cline, Daren B. H.; Samorodnitsky, Gennady

doi:10.1016/0304-4149(94)90113-9

Cited by 341 publications

(185 citation statements)

References 24 publications

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“…Now we prove item (4). The convergence of the series ∞ k=1 1 (w k =w * ) is implied by condition (3.1).…”

Section: (4) Suppose That F ∈ S ∩ R −∞ and Thatmentioning

confidence: 73%

Weighted sums of subexponential random variables and their maxima

Chen

Tang³

2005

Adv. Appl. Probab.

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Let {X k , k = 1, 2, . . .} be a sequence of independent random variables with common subexponential distribution F , and let {w k , k = 1, 2, . . .} be a sequence of positive numbers. Under some mild summability conditions, we establish simple asymptotic estimates for the extreme tail probabilities of the weighted sum n k=1 w k X k and of the maxima of weighted sums max 1≤m≤n m k=1 w k X k , subject to the requirement that they should hold uniformly for n = 1, 2, . . .. A direct application of the result is to risk analysis, where the ruin probability is to be evaluated for a company having the gross loss X k during the kth year with a discount or inflation factor w k .

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“…Now we prove item (4). The convergence of the series ∞ k=1 1 (w k =w * ) is implied by condition (3.1).…”

Section: (4) Suppose That F ∈ S ∩ R −∞ and Thatmentioning

confidence: 73%

Weighted sums of subexponential random variables and their maxima

Chen

Tang³

2005

Adv. Appl. Probab.

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“…Some closely related discussions of the class C can be found in Cline and Samorodnitsky (1994). Recently, Jelenković and Lazar (1999) used the regularity property in (2.1) when they considered some problems in queueing systems and applications.…”

Section: Heavy-tailed Distributionsmentioning

confidence: 99%

Precise large deviations for sums of random variables with consistently varying tails

Tang

Yan

et al. 2004

J. Appl. Probab.

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Let {X k , k ≥ 1} be a sequence of independent, identically distributed nonnegative random variables with common distribution function F and finite expectation µ > 0. Under the assumption that the tail probability F (x) = 1 − F (x) is consistently varying as x tends to infinity, this paper investigates precise large deviations for both the partial sums S n and the random sums S N (t) , where N (t) is a counting process independent of the sequence {X k , k ≥ 1}. The obtained results improve some related classical ones. Applications to a risk model with negatively associated claim occurrences and to a risk model with a doubly stochastic arrival process (extended Cox process) are proposed.

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“…Hence one obtains for ths compound process ratio, 56) which is equivalent to the asymptotic equivalence statement that…”

Section: Lemma 234 (Dominated Convergence Theorem)mentioning

confidence: 99%

“…The following relationship between Matuszewske indexs is known for positive functions f Of direct relevance to the results to be discussed in this manuscript on higher order asymptotic tail expansions will be the extension of the concept of Matuszewska indices which was further developed for distribution function tails in [56] who provided the statements in Lemma 2.26.…”

Section: Definition 224 (Matuszewska Index)mentioning

confidence: 99%

Understanding Operational Risk Capital Approximations: First and Second Orders

2013

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We set the context for capital approximation within the framework of the Basel II / III regulatory capital accords. This is particularly topical as the Basel III accord is shortly due to take effect. In this regard, we provide a summary of the role of capital adequacy in the new accord, highlighting along the way the significant loss events that have been attributed to the Operational Risk class that was introduced in the Basel II and III accords. Then we provide a semi-tutorial discussion on the modelling aspects of capital estimation under a Loss Distributional Approach (LDA). Our emphasis is to focuss on the important loss processes with regard to those that contribute most to capital, the so called "high consequence, low frequency" loss processes.This leads us to provide a tutorial overview of heavy tailed loss process modelling in OpRisk under Basel III, with discussion on the implications of such tail assumptions for the severity model in an LDA structure. This provides practitioners with a clear understanding of the features that they may wish to consider when developing OpRisk severity models in practice.From this discussion on heavy tailed severity models, we then develop an understanding of the impact such models have on the right tail asymptotics of the compound loss process and we provide detailed presentation of what are known as first and second order tail approximations for the resulting heavy tailed loss process. From this we develop a tutorial on three key families of risk measures and their equivalent second order asymptotic approximations: Value-at-Risk (Basel III industry standard); Expected Shortfall (ES) and the Spectral Risk Measure. These then form the capital approximations.We then provide a few example case studies to illustrate the accuracy of these asymptotic captial approximations, the rate of the convergence of the assymptotic result as a function of the LDA frequency and severity model parameters, the sensitivity of the capital approximation to the model parameters and the sensitivity to model miss-specification.Key words: Basel II/III; Capital Approximation; Loss Distributional Approach; Capital Approximation; Value-at-Risk; Expected Shortfall; Spectral Risk Measure; Subexponential; Regularly Varying. The Changing Landscape of Capital AccordsIn juristictions in which active regulation is applied to the banking sector throughout the world, the modelling of Operational Risk (OpRisk) has progressively taken a prominent place in financial quantitative measurement. This has occurred as a result of Basel II and now Basel III regulatory requirements. For example in the context of banking regulation in Australia, the basic framework of Basel II/III is summarized in Figure 1. In this juristiction one observes that a large amount of the developments in quantitative methodology for estimation of OpRisk capital, development of OpRisk frame- There have been both numerical and simulation based approaches adopted as well as analytical mathematical developments of closed form approximatio...

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Subexponentiality of the product of independent random variables

Cited by 341 publications

References 24 publications

Weighted sums of subexponential random variables and their maxima

Weighted sums of subexponential random variables and their maxima

Precise large deviations for sums of random variables with consistently varying tails

Understanding Operational Risk Capital Approximations: First and Second Orders

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