Approximation of the tail probability of randomly weighted sums and applications

Zhang, Yi; Shen, Xinmei; Weng, Chengguo

doi:10.1016/j.spa.2008.03.004

Cited by 98 publications

(68 citation statements)

References 18 publications

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“…A recent new trend of study is to introduce various dependence structures to describe the insurance and financial risks {X i , i ≥ 1} and {Y i , i ≥ 1}. One trend is to require the insurance risks {X i , i ≥ 1} to obey a certain dependence structure (see [4][5][6] among others). Another trend is to assume that {(X i , Y i ), i ≥ 1} form a sequence of i.i.d.…”

Section: Introductionmentioning

confidence: 99%

Ruin Probabilities in a Dependent Discrete-Time Risk Model With Gamma-Like Tailed Insurance Risks

Huang

Zhang

Yang

et al. 2017

Risks

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This paper considered a dependent discrete-time risk model, in which the insurance risks are represented by a sequence of independent and identically distributed real-valued random variables with a common Gamma-like tailed distribution; the financial risks are denoted by another sequence of independent and identically distributed positive random variables with a finite upper endpoint, but a general dependence structure exists between each pair of the insurance risks and the financial risks. Following the works of Yang and Yuen in 2016, we derive some asymptotic relations for the finite-time and infinite-time ruin probabilities. As a complement, we demonstrate our obtained result through a Crude Monte Carlo (CMC) simulation with asymptotics.

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Section: Introductionmentioning

confidence: 99%

Ruin Probabilities in a Dependent Discrete-Time Risk Model With Gamma-Like Tailed Insurance Risks

Huang

Zhang

Yang

et al. 2017

Risks

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“…Note that, we require the upper endpoint of the support of X to be ∞. In recent times, there have been quite a few articles devoted to the asymptotic tail behavior of randomly weighted sums and their maxima, (see, for example, Chen et al, 2005, Hult and Samorodnitsky, 2008, Resnick and Willekens, 1991, Wang and Tang, 2006, Zhang et al, 2009.…”

Section: Introductionmentioning

confidence: 99%

“…The case, when Θ t 's are random, arises in various areas, especially in actuarial and economic situations and stochastic recurrence equation. For various applications, see Hult and Samorodnitsky (2008), Zhang et al (2009). Resnick and Willekens (1991) showed that if {X t } is a sequence of i.i.d.…”

Section: Introductionmentioning

confidence: 99%

“…nonnegative random variables with regularly varying tail of index −α, where α > 0 and {Θ t } is another sequence of positive random variables independent of {X t }, the series X (∞) has regularly varying tail under the following conditions, which we shall call the RW conditions: In particular, if 0 < α < 1, choose ǫ ′ < ǫ such that α + ǫ ′ < 1. Note that Zhang et al (2009) considered the tails of n t=1 Θ t X t and the tails of their maxima, when {X t } are pairwise asymptotically independent and have extended regularly varying and negligible left tail and {Θ t } are positive random variables independent of {X t }. The sufficient conditions for the tails to be regularly varying are almost similar.…”

Section: Introductionmentioning

confidence: 99%

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Tail Behavior of Randomly Weighted Sums

Hazra

Maulik

2012

Adv. Appl. Probab.

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Abstract. Let {Xt, t ≥ 1} be a sequence of identically distributed and pairwise asymptotically independent random variables with regularly varying tails and {Θt, t ≥ 1} be a sequence of positive random variables independent of the sequence {Xt, t ≥ 1}. We shall discuss the tail probabilities and almost sure convergence of X (∞) = ∞ t=1 ΘtX + t (where X + = max{0, X}) and max 1≤k<∞ k t=1 ΘtXt and provide some sufficient conditions motivated by Denisov and Zwart (2007) as alternatives to the usual moment conditions. In particular, we illustrate how the conditions on the slowly varying function involved in the tail probability of X 1 helps to control the tail behavior of the randomly weighted sums.Note that, the above results allow us to choose X 1 , X 2 , . . . as independent and identically distributed positive random variables. If X 1 has regularly varying tail of index −α, where α > 0, and if {Θt, t ≥ 1} is a positive sequence of random variables independent of {Xt}, then it is known, which can also be obtained from the sufficient conditions above, that under some appropriate moment conditions on {Θt, t ≥ 1}, X (∞) = ∞ t=1 ΘtXt converges with probability 1 and has regularly varying tail of index −α. Motivated by the converse problems in Jacobsen et al. (2009) we ask the question that, if X (∞) has regularly varying tail, then does X 1 have regularly varying tail under some appropriate conditions? We obtain appropriate sufficient moment conditions, including nonvanishing Mellin transform of ∞ t=1 Θt along some vertical line in the complex plane, so that the above is true. We also show that the condition on the Mellin transform cannot be dropped.

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“…A recent new approach is to introduce various dependence structures to the risk model. In this direction, we refer the reader to Goovaerts et al (2005), Tang (2006a), Zhang et al (2009), Weng et al (2009), and Yi et al (2011), among many others. However, there are few papers which take into account the dependence between insurance and financial risks, with the difficulty existing in describing the tail behavior of the product of dependent random variables.…”

Section: Introductionmentioning

confidence: 99%

The Finite-Time Ruin Probability with Dependent Insurance and Financial Risks

Chen

2011

J. Appl. Probab.

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Consider a discrete-time insurance risk model. Within period i, the net insurance loss is denoted by a real-valued random variable X i . The insurer makes both risk-free and risky investments, leading to an overall stochastic discount factor Y i from time i to time i − 1. Assume that (X i , Y i ), i ∈ N, form a sequence of independent and identically distributed random pairs following a common bivariate Farlie-Gumbel-Morgenstern distribution with marginal distribution functions F and G. When F is subexponential and G fulfills some constraints in order for the product convolution of F and G to be subexponential too, we derive a general asymptotic formula for the finite-time ruin probability. Then, for special cases in which F belongs to the Fréchet or Weibull maximum domain of attraction, we improve this general formula to be transparent.

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Approximation of the tail probability of randomly weighted sums and applications

Cited by 98 publications

References 18 publications

Ruin Probabilities in a Dependent Discrete-Time Risk Model With Gamma-Like Tailed Insurance Risks

Ruin Probabilities in a Dependent Discrete-Time Risk Model With Gamma-Like Tailed Insurance Risks

Tail Behavior of Randomly Weighted Sums

The Finite-Time Ruin Probability with Dependent Insurance and Financial Risks

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