Tail asymptotics for the sum of two heavy-tailed dependent risks

Let X ₁ , X ₂ denote positive heavy-tailed random variables with continuous marginal distribution functions F ₁ and F ₂, respectively. The asymptotic behavior of the tail of X ₁ +X ₂ is studied in a general copula framework and some bounds and extremal properties are provided. For more specific assumptions on F ₁ , F ₂ and the underlying dependence structure of X ₁ and X ₂, we survey explicit asymptotic results available in the literature and add several new cases.Supported by the Austrian Science Fund Project P-18392.     

On exact rates of decay of solutions of linear systems of Volterra equations with delay

This paper considers the resolvent of a finite-dimensional linear convolution Volterra integral equation. The main results give conditions which ensure that the exact rate of decay of the resolvent can be determined using a positive weight function related to the kernel. The decay rates can be exponential or subexponential. Many other related results on exact rates of exponential and subexponential decay of solutions of Volterra integro-differential equations are given. We also present an application to a linear compartmental system with discrete and continuous lags.     

On upper bounds for the tail distribution of geometric sums of subexponential random variables

The approach used by Kalashnikov and Tsitsiashvili for constructing upper bounds for the tail distribution of a geometric sum with subexponential summands is reconsidered. By expressing the problem in a more probabilistic light, several improvements and one correction are made, which enables the constructed bound to be significantly tighter. Several examples are given, showing how to implement the theoretical result.     

Characterizations of counter-monotonicity and upper comonotonicity by (tail) convex order

In this paper, we characterize counter-monotonic and upper comonotonic random vectors by the optimality of the sum of their components in the senses of the convex order and tail convex order respectively. In the first part, we extend the characterization of comonotonicity by  Cheung (2010) and show that the sum of two random variables is minimal with respect to the convex order if and only if they are counter-monotonic. Three simple and illuminating proofs are provided. In the second part, we investigate upper comonotonicity by means of the tail convex order. By establishing some useful properties of this relatively new stochastic order, we prove that an upper comonotonic random vector must give rise to the maximal tail convex sum, thereby completing the gap in  Nam et al. (2011)’s characterization. The relationship between the tail convex order and risk measures along with conditions under which the additivity of risk measures is sufficient for upper comonotonicity is also explored.     

Approximation and estimation of some compound distributions

The distribution of the total amount claimed up to time t can often be written in the form of a compound distribution G_t(x) = Σp_n(t)F⁽ⁿ⁾(x) where p_n(t) is the probability of exactly n claims while F is the distribution of a single claim. In the actuarial literature one often finds approximations of G_t(x) when the time t is large. It seems more natural to take t fixed and to look for approximations for x large. This paper contains a number of such results for a Poisson process and for a Pascal process. Different hypotheses on the tail behaviour of F(t) yield different expressions to estimate 1 - G_t(x). The results obtained should prove to have wider applicability than suggested by the insurance context. Within it, however, applications to premium calculation principles are immediate.     

On selecting an extreme value distribution

In a recent paper Hernández and Johnson (1984) have given a procedure based on Bayesian statistical inference for selecting an extreme-value distribution to best fit available data. In this note we give an alternative derivation of part of their results. This derivation — based onstructural inference (Fraser 1968) — provides theoretical support for these results that would not be present on the basis of their derivation using Bayesian techniques.Supported in part by a Washington State University Faculty Summer Support Award 1984.     

On two supercongruences of double binomial sums

Periodica Mathematica Hungarica - In this note we confirm two conjectural supercongruences of double sums of binomial coefficients due to El Bachraoui.     

负相依随机变量的自正则化中心极限定理与部分和的方差估计

§ 1　 IntroductionWe firstintroduce some concepts.Random variables X and Y are called negative dependent ( ND) if for any pair ofmonotonically non-decresing functions f and g,Cov{ f( X) ,g( Y) }≤ 0 .Clearly itis equivalenttoP( X≤ x,Y≤ y)≤ P( X≤ x) P( Y≤ y)for all x,y∈R.A random sequence{ Xi,i≥ 1 } is said to be negative quadrant dependent( NQD) if any pairof variables Xi,Xj( i≠j) are ND.A sequence of random variables{ Xi,i≥ 1 } is said to be linear negative quadrand depend…     

Bounds and approximations for sums of dependent log-elliptical random variables

Dhaene, Denuit, Goovaerts, Kaas and Vyncke [Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R., Vyncke, D., 2002a. The concept of comonotonicity in actuarial science and finance: theory. Insurance Math. Econom. 31 (1), 3-33; Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R., Vyncke, D., 2002b. The concept of comonotonicity in actuarial science and finance: Applications. Insurance Math. Econom. 31 (2), 133-161] have studied convex bounds for a sum of dependent random variables and applied these to sums of log-normal random variables. In particular, they have shown how these convex bounds can be used to derive closed-form approximations for several of the risk measures of such a sum. In this paper we investigate to which extent their general results on convex bounds can also be applied to sums of log-elliptical random variables which incorporate sums of log-normals as a special case. Firstly, we show that unlike the log-normal case, for general sums of log-ellipticals the convex lower bound does no longer result in closed-form approximations for the different risk measures. Secondly, we demonstrate how instead the weaker stop-loss order can be used to derive such closed-form approximations. We also present numerical examples to show the accuracy of the proposed approximations.     

On limit laws for sums of Pfeifer records

The problem of determining limiting distributions for sums of records has been studied by several authors who have considered a variety of assumptions sufficient to ensure that sums of records properly normalized will converge to a non-degenerate distribution. As a parallel to these endeavors, it is of interest to establish conditions under which the sum of Pfeifer records, properly normalized, converges. Pfeifer records are defined under the assumption that initial observations are i.i.d. with common survival function and following the (n−1)-th record value the observations are assumed to have survival function ,n=1,2,.... The study of the asymptotic behavior of sums of Pfeifer records constitutes a natural generalization of work on sums of classical records. The present paper introduces conditions under which the limit distribution of sums of Pfeifer records is non-degenerate.      

投资组合风险的分散化研究

风险是金融投资领域的研究热点问题之下一,投资组合是降低投资风险的有效方法之一。人们在做出投资决策时总是追求在一定收益率下风险最小。本文论述了投资组合收益和风险的数学统计方法,阐明风险可分为系统风险和非系统风险,后者可以通过投资组合分散化。本文还探讨了证券相关性和组合风险之间的关系。最后作了实证分析。     

随机变量的负超可加相依及其应用

一个随机向量Ｘ＝（Ｘ１,Ｘ２,．．．,Ｘｍ）称为负超可加相依（ＮＳＤ）,如果对每个超可加函数Х,ＥХ（Ｘ１,Ｘ２,．．．,Ｘｍ）≥ＥХ（Ｙ１,Ｙ２,．．．,Ｙｍ）,其中Ｙ１,Ｙ２,．．．,Ｙｍ相互独立且对任意ｉ,Ｙｉ＝ｄＸｉ,本文研究了ＮＳＤ的基本性质,给出了ＮＳＤ判定的三个结构 定理,并且这些定理可用来证明许多著名的多元分布具有ＮＳＤ性质,本文还给出了ＮＡＤ的许多概率不等式。     

On the structure of rings which are sums of two subrings

We study the structure of rings that are sums of two subrings one of which is nil and the other reduced. Our main results concern the problem whether in this situation the nil subring must be an ideal.Received: 13 July 2001     

Complete convergence for weighted sums of LNQD random variables

In this paper, we study the complete convergence for weighted sums of linearly negative quadrant dependent (LNQD) random variables based on the exponential bounds. In addition, we present some complete convergence for arrays of rowwise LNQD random variables.