Weighted sums of subexponential random variables and asymptotic dependence between returns on reinsurance equities

Abstract

Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. ABSTRACT. Suppose X 1 , X 2 , . . . are independent subexponential random variables with partial sums S n . We show that if the pairwise sums of the X i 's are subexponential, then S n is subexponential and Terms of use: Documents inThe result is applied to give conditions under which P (Asymptotic tail probabilities for bivariate linear combinations of subexponential random variables are given. These results are applied to explain the joint movements of the stocks of reinsurers. Portfolio investment and retrocession practices in the reinsurance industry, for reasons of diversification, exposes different reinsurers to the same risks on both sides of their balance sheets. Assuming, in line with the industry practice that the risk drivers follow subexponential distributions, we derive (under mild conditions) when the reinsurer's equity returns are asymptotically dependent, exposing the industry to systemic risk.

Asymptotic expansions for infinite weighted convolutions of rapidly varying subexponential distributions

“…We remark that Geluk and De Vries [17] gives a different formulation of the above result; see also Chen et al [7].…”

“…Examples of applications of subexponentiality include transient renewal theory [14,31], random walks [19,32], branching processes [2,8,9], queueing theory [24], shot noise [21], infinite divisibility [15], ruin theory [1,18,29,30], compound sums [10,13,20], insurance risk theory [16], and heavy-tailed linear processes [7,11,17,28]. In these papers, inputs to the processes follow a subexponential distribution and an asymptotic analysis of an output is obtained, e.g., claim size distribution and ruin probability, age distribution and expected number of particles alive, service time distribution and stationary waiting time distribution, or innovation distribution and tail area for weighted averages.…”

Asymptotic Tail Probabilities of Sums of Dependent Subexponential Random Variables

“…Scattered discussions at this point can be found in Ng et al [25], Asmussen et al [3], Geluk and De Vries [14], and Foss et al [11], among others. We remark that the assumption of independence among the underlying random variables appears far too unrealistic in most practical situations and it considerably limits the usefulness of obtained results.…”

On the Distribution of the Nearly Unstable AR(1) Process with Heavy Tails

“…In [16] a negative drift random walk with dependent step sizes given by a two-sided linear process with regularly varying innovations was considered, and, as part of their analysis, close to minimal conditions under which (2.5) holds were derived, including the case in which α ≥ 1. First-and second-order asymptotics for infinite weighted sums with regularly varying tails were given in [1], and generalizations to more general subexponential distributions were given in [14]. Finally, partial weighted sums and their maxima, where the increments belong to a large family of subexponential distributions, were considered in [7].…”