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15
Sums of dependent nonnegative random variables with subexponential tails
 Journal of Applied Probability
, 2008
"... In this paper we study the asymptotic tail probabilities of sums of subexponential nonnegative random variables, which are dependent according to certain general structures with tail independence. The results show that the subexponentiality of the summands eliminates the impact of the dependence on ..."
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Cited by 12 (2 self)
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In this paper we study the asymptotic tail probabilities of sums of subexponential nonnegative random variables, which are dependent according to certain general structures with tail independence. The results show that the subexponentiality of the summands eliminates the impact of the dependence on the tail behavior of the sums.
Sums of pairwise quasiasymptotically independent random variables with consistent variation
 Stochastic Models
, 2009
"... This article investigates the tail asymptotic behavior of the sum of pairwise quasiasymptotically independent random variables with consistently varying tails. We prove that the tail probability of the sum is asymptotically equal to the sum of individual tail probabilities. This matches a feature o ..."
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Cited by 11 (1 self)
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This article investigates the tail asymptotic behavior of the sum of pairwise quasiasymptotically independent random variables with consistently varying tails. We prove that the tail probability of the sum is asymptotically equal to the sum of individual tail probabilities. This matches a feature of subexponential distributions. This result is then extended to weighted sums and random sums.
Asymptotics for weighted random sums
 Advances in Applied Probability 44 (2012
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Heavy Tails of Discounted Aggregate Claims in the Continuoustime Renewal Model
, 2007
"... We study the tail behavior of discounted aggregate claims in a continuoustime renewal model. For the case of Paretotype claims, we establish a tail asymptotic formula, which holds uniformly in time. ..."
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Cited by 5 (1 self)
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We study the tail behavior of discounted aggregate claims in a continuoustime renewal model. For the case of Paretotype claims, we establish a tail asymptotic formula, which holds uniformly in time.
A uniform asymptotic estimate for discounted aggregate claims with subexponential tails
 Insurance Math. Econom
"... In this paper we study the tail probability of discounted aggregate claims in a continuoustime renewal model. For the case that the common claimsize distribution is subexponential, we obtain an asymptotic formula, which holds uniformly for all time horizons within a finite interval. Then, with som ..."
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Cited by 5 (1 self)
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In this paper we study the tail probability of discounted aggregate claims in a continuoustime renewal model. For the case that the common claimsize distribution is subexponential, we obtain an asymptotic formula, which holds uniformly for all time horizons within a finite interval. Then, with some additional mild assumptions on the distributions of the claim sizes and interarrival times, we further prove that this formula holds uniformly for all time horizons. In this way, we significantly extend a recent result of Tang [Tang, Q., 2007. Heavy tails of discounted aggregate claims in the continuoustime renewal model. Journal of Applied Probability 44 (2), 285–294].
A Hybrid Estimate for the Finitetime Ruin Probability in a Bivariate Autoregressive Risk Model with Application to Portfolio Optimization
, 2011
"... Consider a discretetime risk model in which the insurer is allowed to invest a proportion of its wealth in a risky stock and keep the rest in a riskfree bond. Assume that the claim amounts within individual periods follow an autoregressive process with heavytailed innovations and that the logret ..."
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Cited by 2 (2 self)
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Consider a discretetime risk model in which the insurer is allowed to invest a proportion of its wealth in a risky stock and keep the rest in a riskfree bond. Assume that the claim amounts within individual periods follow an autoregressive process with heavytailed innovations and that the logreturns of the stock follow another autoregressive process, independent of the former one. We derive an asymptotic formula for the finitetime ruin probability and propose a hybrid method, combining simulation with asymptotics, to compute this ruin probability more effi ciently. As an application, we consider a portfolio optimization problem in which we determine the proportion invested in the risky stock that maximizes the expected terminal wealth subject to a constraint on the ruin probability.
Subexponential Tails of Discounted Aggregate Claims in a TimeDependent Renewal Risk Model
, 2010
"... Consider a continuoustime renewal risk model with a constant force of interest. We assume that claim sizes and interarrival times correspondingly form a sequence of independent and identically distributed random pairs and that each pair obeys a dependence structure described via the conditional ta ..."
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Consider a continuoustime renewal risk model with a constant force of interest. We assume that claim sizes and interarrival times correspondingly form a sequence of independent and identically distributed random pairs and that each pair obeys a dependence structure described via the conditional tail probability of a claim size given the interarrival time before the claim. We focus on determination of the impact of this dependence structure on the asymptotic tail probability of discounted aggregate claims. Assuming that the claimsize distribution is subexponential, we derive an exact locally uniform asymptotic formula, which quantitatively captures the impact of the dependence structure. When the claimsize distribution is extendedregularlyvarying tailed, we show that this asymptotic formula is globally uniform.
Rare Event Simulation for Linear Combinations of LogElliptical Random Variables
, 2014
"... The probability that a sum of lognormal random variables exceeds some value u is of great interest in areas such as insurance and finance. There exists a conditional Monte Carlo estimator that allows these probabilities to be computed efficiently. Here we show that by extending the problem to permi ..."
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The probability that a sum of lognormal random variables exceeds some value u is of great interest in areas such as insurance and finance. There exists a conditional Monte Carlo estimator that allows these probabilities to be computed efficiently. Here we show that by extending the problem to permit general linear combinations of logelliptical random variables, we lose the properties that the estimator required to function. A new conditional Monte Carlo estimator is proposed to handle the challenges that arise from allowing general linear combinations. The algorithm used to compute the estimator introduces a new root finding technique designed to find all roots of a function within a given interval. In addition, we vary the input to the general linear combination and analyse the results to discover focus points. These are inputs which give high probability that the general linear combination exceeds u. This leads to the proposal of an importance sampling scheme for the new estimator. The importance sampling is based upon a mixture of vonMises Fisher distributions, each centred around a focus point. Developing a new conditional Monte Carlo estimator that handles general linear combinations of lognormal random variables allows us to solve the problems that would previously be too difficult to solve accurately. i
The Maximum of Randomly Weighted Sums with Long Tails in Insurance and Finance
, 2011
"... In risk theory we often encounter stochastic models containing randomly weighted sums. In these sums, each primary realvalued random variable, interpreted as the net loss during a reference period, is associated with a nonnegative random weight, interpreted as the corresponding stochastic discount ..."
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In risk theory we often encounter stochastic models containing randomly weighted sums. In these sums, each primary realvalued random variable, interpreted as the net loss during a reference period, is associated with a nonnegative random weight, interpreted as the corresponding stochastic discount factor to the origin. Therefore, a weighted sum of m terms, denoted as S (w) m, represents the stochastic present value of aggregate net losses during the first m periods. Suppose that the primary random variables are independent of each other with longtailed distributions and are independent of the random weights. We show conditions on the random weights under which the tail probability of max1≤m≤n S (w) m the maximum of the first n weighted sums is asymptotically equivalent to that of S (w) n the last weighted sum.
On the Maximum of Randomly Weighted Sums with Regularly Varying Tails
, 2005
"... Consider the randomly weighted sums Sn(θ) = ∑n k=1 θkXk, n = 1, 2,..., where {Xk, k = 1, 2,...} is a sequence of independent realvalued random variables with common distribution F, whose right tail is regularly varying with exponent −α < 0, and {θk, k = 1, 2,...} is a sequence of positive random ..."
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Consider the randomly weighted sums Sn(θ) = ∑n k=1 θkXk, n = 1, 2,..., where {Xk, k = 1, 2,...} is a sequence of independent realvalued random variables with common distribution F, whose right tail is regularly varying with exponent −α < 0, and {θk, k = 1, 2,...} is a sequence of positive random variables, independent of {Xk, k = 1, 2,...}. Under a suitable summability condition on the upper endpoints of θk, k = 1, 2,..., we prove that Pr (max1≤n< ∞ Sn(θ)> x) ∼ F (x) k=1 Eθ