Asymptotics in the symmetrization inequality

We give a sufficient condition for i.i.d. random variables X1,X2 in order to have P{X1-X2>x} ~ P{|X1|>x} as x tends to infinity. A factorization property for subexponential distributions is used in the proof. In a subsequent paper the results will be applied to model fragility of financial markets

Tails of subordinated laws: The regularly varying case

Suppose Xi, i = 1,2,... are i.i.d. positive random variables with d.f. F. We assume the tail d.f. F̄ = 1 - F to be regularly varying (F̄(tx)|F̄(t) → x-β,x > 0,t → ∞) with 0 x) as x → ∞ where SN = ΣN 1 Xi and N,Xi(i≥ 1) independent with Σ∞ n=0P(N = n)xn analytic at x = 1 is studied under an additional smoothness condition on F. As an application we give the asymptotic behaviour of the expected population size of an age-dependent branching process

Renewal theory for random variables with a heavy tailed distribution and finite variance

Let X-1, X-2,... X-n be independent and identically distributed (i.i.d.) non-negative random variables with a common distribution function (d.f.) F with unbounded support and EX12 < infinity. We show that for a large class of heavy tailed random variables with a finite variance the renewal function U satisfies U(x) - x/mu - mu(2)/2 mu(2) similar to -1/mu x integral(infinity)(x) integral(infinity)(s) (1 - F(u))duds as x -> infinity

Renewal theory for random variables with a heavy tailed distribution and finite variance

In this paper we show for a large class of heavy tailed random variables a second order asymptotic result for the well-known renewal functio

On bootstrap sample size in extreme value theory

It has been known for a long time that for bootstrapping the probability distribution of the maximum of a sample consistently, the bootstrap sample size needs to be of smaller order than the original sample size. See Jun Shao and Dongsheng Tu (1995), Ex. 3.9,p. 123. We show that the same is true if we use the bootstrap for estimating an intermediate quantile

Weighted Sums of Subexponential Random Variables and Asymptotic Dependence between Returns on Reinsurance Equities

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

Convolutions of heavy-tailed random variables and applications to portfolio diversification and MA(1) time series

The paper characterizes first and second order tail behavior of convolutions of i.i.d. heavy tailed random variables with support on the real line. The result is applied to the problem of risk diversification in portfolio analysis and to the estimation of the parameter in a MA(1) model

Weak & Strong Financial Fragility

The stability of the financial system at higher loss levels is either characterized by asymptotic dependence or asymptotic independence. If asymptotically independent, the dependency, when present, eventually dies out completely at the more extreme quantiles, as in case of the multivariate normal distribution. Given that financial service firms' equity returns depend linearly on the risk drivers, we show that the marginals' distributions maximum domain of attraction determines the type of systemic (in-)stability. A scale for the amount of dependency at high loss lovels is designed. This permits a characterization of systemic risk inherent to different financial network structures. The theory also suggests the functional form of the economically relevant limit copulas

Stable Probability Distributions and their Domains of Attraction

The theory of stable probability distributions and their domains of attraction is derived in a direct way (avoiding the usual route via infinitely divisible distributions) using Fourier transforms. Regularly varying functions play an important role in the exposition