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Subexponential Distributions
, 1997
"... We survey the properties and uses of the class of subexponential probability distributions, paying particular attention to their use in modelling heavytailed data such as occurs in insurance and queueing applications. We give a detailed summary of the core theory and discuss subexponentiality in va ..."
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Cited by 38 (7 self)
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We survey the properties and uses of the class of subexponential probability distributions, paying particular attention to their use in modelling heavytailed data such as occurs in insurance and queueing applications. We give a detailed summary of the core theory and discuss subexponentiality in various contexts including extremes, random walks and L'evy processes with negative drift, and sums of random variables, the latter extended to cover random sums, weighted sums and moving averages. 1. Definition and first properties Subexponential distributions are a special class of heavytailed distributions. The name arises from one of their properties, that their tails decrease more slowly than any exponential tail; see (1.4). This implies that large values can occur in a sample with nonnegligible probability, and makes the subexponential distributions candidates for modelling situations where some extremely large values occur in a sample compared to the mean size of the data. Such a p...
Asymptotic results for the sum of dependent nonidentically distributed random variables, Methodology and Computing
 in Applied Probability. Forthcoming. DOI
, 2008
"... In this paper we extend some results about the probability that the sum of n dependent subexponential random variables exceeds a given threshold u. In particular, the case of nonidentically distributed and not necessarily positive random variables is investigated. Furthermore we establish criteria ..."
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Cited by 25 (1 self)
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In this paper we extend some results about the probability that the sum of n dependent subexponential random variables exceeds a given threshold u. In particular, the case of nonidentically distributed and not necessarily positive random variables is investigated. Furthermore we establish criteria how far the tail of the marginal distribution of an individual summand may deviate from the others so that it still influences the asymptotic behavior of the sum. Finally we explicitly construct a dependence structure for which, even for regularly varying marginal distributions, no asymptotic limit of the tail sum exists. Some explicit calculations for diagonal copulas and tcopulas are given. 1
Delay bounds in communication networks with heavytailed and selfsimilar traffic
 IEEE Transactions on Information Theory
, 2012
"... Traffic with selfsimilar and heavytailed characteristics has been widely reported in communication networks, yet, the stateoftheart of analytically predicting the delay performance of such networks is lacking. We address a particularly difficult type of heavytailed traffic where only the first ..."
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Cited by 12 (3 self)
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Traffic with selfsimilar and heavytailed characteristics has been widely reported in communication networks, yet, the stateoftheart of analytically predicting the delay performance of such networks is lacking. We address a particularly difficult type of heavytailed traffic where only the first moment can be computed, and present nonasymptotic endtoend delay bounds for such traffic. The derived performance bounds are nonasymptotic in that they do not assume a steady state, large buffer, or many sources regime. The analysis follows a network calculus approach where traffic is characterized by envelope functions and service is described by service curves. Our analysis considers a multihop path of fixedcapacity links with heavytailed selfsimilar cross traffic at each node. A key contribution of the analysis is a novel probabilistic samplepath bound for heavytailed arrival and service processes, which is based on a scalefree sampling method. We explore how delays scale as a function of the length of the path, and compare them with lower bounds. A comparison with simulations illustrates pitfalls when simulating selfsimilar heavytailed traffic, providing further evidence for the need of analytical bounds. I.
Pricing catastrophe insurance products based on actually reported claims
, 1998
"... Abstract This paper deals with the problem of pricing a financial product relying on an index of reported claims from catastrophe insurance. The problem of pricing such products is that, at a fixed time in the trading period, the total claim amount from the catastrophes occurred is not known. There ..."
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Cited by 10 (1 self)
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Abstract This paper deals with the problem of pricing a financial product relying on an index of reported claims from catastrophe insurance. The problem of pricing such products is that, at a fixed time in the trading period, the total claim amount from the catastrophes occurred is not known. Therefore, one has to price these products solely from knowing the aggregate amount of the reported claims at the fixed time point. This paper will propose a way to handle this problem, and will thereby extend the existing pricing models for products of this kind.
Large deviations for heavytailed random sums in compound renewal
"... Abstract In the present paper we investigate the precise large deviations for heavytailed random sums. First, we obtain a result which improves the relative result in Kl uppelberg and Mikosch (J. Appl. Probab. 34 (1997) 293). Then we introduce a more realistic risk model than classical ones, named ..."
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Cited by 8 (1 self)
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Abstract In the present paper we investigate the precise large deviations for heavytailed random sums. First, we obtain a result which improves the relative result in Kl uppelberg and Mikosch (J. Appl. Probab. 34 (1997) 293). Then we introduce a more realistic risk model than classical ones, named the compound renewal model, and establish the precise large deviations in this model. c 2001 Elsevier Science B.V. All rights reserved MSC: primary 60F10; 60F05; 60G50; secondary 60K10; 62P05
Pricing Insurance Derivatives, the Case of CATFutures
 Risk, George State University Atlanta, Society of Actuaries, Monograph
, 1997
"... Since their appearance on the market, catastrophe insurance futures have triggered a considerable interest from both practitioners as well as academics. As one example of a securitized (re)insurance risk, its pricing and hedging contains many of the key problems to be addressed in the analysis of mo ..."
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Cited by 8 (1 self)
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Since their appearance on the market, catastrophe insurance futures have triggered a considerable interest from both practitioners as well as academics. As one example of a securitized (re)insurance risk, its pricing and hedging contains many of the key problems to be addressed in the analysis of more general insurance derivatives. In the present paper we review the main methodological questions underlying the theoretical pricing of such products. We discuss utility maximization pricing more in detail. A key methodological feature is the theory of incomplete markets. Our paper follows closely the exposition given in Meister (1995). Catastrophe Insurance Futures
RUIN PROBABILITIES FOR RISK PROCESSES WITH NONSTATIONARY ARRIVALS AND SUBEXPONENTIAL CLAIMS
"... Abstract. In this paper, we obtain the finitehorizon and infinitehorizon ruin probability asymptotics for risk processes with claims of subexponential tails for nonstationary arrival processes that satisfy a large deviation principle. As a result, the arrival process can be dependent, nonstation ..."
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Cited by 6 (6 self)
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Abstract. In this paper, we obtain the finitehorizon and infinitehorizon ruin probability asymptotics for risk processes with claims of subexponential tails for nonstationary arrival processes that satisfy a large deviation principle. As a result, the arrival process can be dependent, nonstationary and nonrenewal. We give three examples of nonstationary and nonrenewal point processes: Hawkes process, Cox process with shot noise intensity and selfcorrecting point process. We also show some aggregate claims results for these three examples. Let us consider a classical risk model
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.
Approximating the moments of marginals of high dimensional distributions, Annals of Probability, to appear
 Department of Mathematics, University of Michigan
"... For probability distributions on R n, we study the optimal sample size N = N(n, p) that suffices to uniformly approximate the pth moments of all onedimensional marginals. Under the assumption that the marginals have bounded 4p moments, we obtain the optimal bound N = O(n p/2) for p>2. This bound ..."
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Cited by 5 (1 self)
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For probability distributions on R n, we study the optimal sample size N = N(n, p) that suffices to uniformly approximate the pth moments of all onedimensional marginals. Under the assumption that the marginals have bounded 4p moments, we obtain the optimal bound N = O(n p/2) for p>2. This bound goes in the direction of bridging the two recent results: a theorem of Guedon and Rudelson [Adv. Math. 208 (2007) 798–823] which has an extra logarithmic factor in the sample size, and a result of Adamczak et al. [J. Amer. Math. Soc. 23 (2010) 535–561] which requires stronger subexponential moment assumptions. 1. Introduction. 1.1. The estimation problem. We study the following problem: how well can one approximate onedimensional marginals of a distribution on Rn by sampling? Consider a random vector X in Rn, and suppose we would like to compute the pth moments of the marginals 〈X, x 〉 for all x ∈ Rn. To this end, we sample N