Dominated variation and related concepts and Tauberian theorems for Laplace transforms

Some Abelian and Tauberian theorems are proved under conditions of dominated variation and related concepts. For example U is dominatedly varying if and only if its Laplace-Stieltjes transform is dominatedly varying.     

宽象限相依随机变量部分和的中偏差及其在保险中的应用

研究了控制变换尾分布的宽象限相依实值随机变量部分和的中偏差.相应于所得到的理论结果,进一步给出了在相依保险风险模型中的两个应用;一是在基于顾客到达过程的保险风险模型中,保险公司盈余的渐近估计;二是在复合更新风险模型中,有限时和无限时破产概率的一致渐近估计.     

Estimates for the ruin probability of a time-dependent renewal risk model with dependent by-claims

Consider a continuous-time renewal risk model, in which every main claim induces a delayed by-claim. Assume that the main claim sizes and the inter-arrival times form a sequence of identically distributed random pairs, with each pair obeying a dependence structure, and so do the by-claim sizes and the delay times. Supposing that the main claim sizes with by-claim sizes form a sequence of dependent random variables with dominatedly varying tails, asymptotic estimates for the ruin probability of the surplus process are investigated, by establishing a weakly asymptotic formula, as the initial surplus tends to infinity.     

Asymptotics for tail probability of total claim amount with negatively dependent claim sizes and its applications

In this paper, we obtain the asymptotics for the tail probability of the total claim amount with negatively dependent claim sizes in two cases: in the first case, the distribution tail of the claim number is dominatedly varying; in the second case, the distribution of the claim number is in the maximum domain of attraction of the Gumbel distribution, and the claim sizes are light-tailed. In both cases, we assume that the claim sizes are nondegenerate negatively dependent and identically distributed random variables and that the claim number is not necessarily independent of the claim sizes. As applications, we derive asymptotics for the finite-time ruin probabilities in some dependent compound renewal risk models with constant interest rate.     

Precise large deviations for widely orthant dependent random variables with dominatedly varying tails

For the widely orthant dependent (WOD) structure, this paper mainly investigates the precise large deviations for the partial sums ofWOD and non-identically distributed random variables with dominatedly varying tails. The obtained results extend some corresponding results.     

负相协D族广义Sparre-Andevsen模型下的大偏差及其应用

本文将经典的Sparre-Andevsen风险模型推广到保费收入过程不再是线性过程的一般风险过程,得到了一些关于负相协D族随机变量随机和的大偏差结果,以及破产概率的弱等价性.     

On sufficient conditions for cost rate function unimodality in periodic review (s, S) inventory models

The cost rate function that arises in the stationary analysis of a class of periodic review regenerative inventory systems is known to be unimodal if the renewal density of the underlying demand sequence is decreasing. We prove that the same result holds, under zero leadtimes, if the renewal density is concave increasing.     

On closure properties of heavy-tailed distributions for random sums*

We investigate some closure properties of heavy-tailed distributions for random sums. We first consider the random sums with independent or some certain dependent long-tailed increments. We also consider the real-valued increments with dominatedly varying-tailed distributions under extended negative dependence and obtain two results on necessary and sufficient conditions or sufficient conditions for the closure on random sums.     

On the ruin probability in a dependent discrete time risk model with insurance and financial risks

This paper considers the discrete-time risk model with insurance risk and financial risk in some dependence structures. Under assumptions that the insurance risks are heavy tailed (belong to the intersection of the long-tailed class and the dominatedly varying-tailed class) and the financial risks satisfy some moment conditions, the asymptotic and uniformly asymptotic relations for the finite-time and ultimate ruin probabilities are derived.     

Capital renewal as a real option

We consider the timing of replacement of obsolete subsystems within an extensive, complex infrastructure. Such replacement action, known as capital renewal, must balance uncertainty about future profitability against uncertainty about future renewal costs. Treating renewal investments as real options, we derive an optimal solution to the infinite horizon version of this problem and determine the total present value of an institution’s capital renewal options. We investigate the sensitivity of the infinite horizon solution to variations in key problem parameters and highlight the system scenarios in which timely renewal activity is most profitable. For finite horizon renewal planning, we show that our solution performs better than a policy of constant periodic renewals if more than two renewal cycles are completed.     

带双阈值的一类更新风险模型的Gerber-Shiu折现罚函数

本文考虑一类索赔时间间隔为Erlang(2)分布的"双界限"分红模型,在这种模型中,当盈余超过上限时分红以不超过保费率的速率付出,低于下限后保费率增大,得到了关于Gerber-Shiu罚金折现期望函数满足的积分-微分方程以及更新方程,进一步讨论了更新方程的解.     

更新随机指标分枝过程的大偏差

研究了时间指标为一般更新过程的随机指标分枝过程.在每个粒子至少有两个分枝(Bottcher情形)以及更新分布满足Cramer条件的情况下,得到了更新随机指标分枝过程的大偏差原理.     

Weibull renewal processes

We study the class of renewal processes with Weibull lifetime distribution from the point of view of the general theory of point processes. We investigate whether a Weibull renewal process can be expressed as a Cox process. It is shown that a Weibull renewal process is a Cox process if and only if 0<1, where denotes the shape parameter of the Weibull distribution. The Cox character of the process is analyzed. It is shown that the directing measure of the process is continuous and singular.     

Reliability of several component sets with inspections at random times

We consider a random process which represents a system of components with constant failure rates and subjected to inspections at times defining a renewal process. We give an analytic method for calculating the reliability function, its Laplace transform and the mean time to failure (MTTF). These formulas are computable if the Laplace transform of the inter-arrival law of the renewal process is explicit. We also study the asymptotic behavior of the reliability and determine the asymptotic failure rate of the system.     

更新过程中的几个随机比较结果

本文研究了更新过程中的几个随机比较结果.通过对不同时刻之前的最后一次更新发生时刻变量在失效率序、反向失效率序和似然比序意义下的随机比较,获得了若底分布F是IFR,则t之前最后一次更新的发生时刻变量SN(t)在失效率序和反向失效率序意义下关于t单调递增;若底分布F是IFR且绝对连续,则SN(t)在似然比序意义下关于t单调递增.     

带有相依重尾冲击的Poisson噪音过程尾的一致渐近性质

本文考虑了带有某种相依重尾冲击的Poisson噪音过程尾的一致渐近性质.当冲击是二元上尾渐近独立的非负随机变量具有长尾和控制变化尾分布且噪音函数具有正的上下界时,得到了过程尾概率的一致渐近公式.进而,当冲击具有连续的一致变化尾分布时,去除了噪音函数具有正的下界的限制.对于噪音函数不一定具有正的上界的情形,当冲击具有两两负象限相依结构时,也得到了一致渐近性结果.     

Total duration of negative surplus for the dual model

In the dual model, we allow the surplus process to continue if the surplus falls below zero. By introducing the renewal measure of the defective renewal sequence constituted by the zero points of the surplus process, we obtain the probability of hitting the zero point. Further, we derive formulae for the Laplace transform, expectation and variance of total duration of negative surplus and present some examples with an exponential individual jump amount distribution. Copyright © 2008 John Wiley & Sons, Ltd.     

Lattice-type self-similar sets with pluriphase generators fail to be Minkowski measurable

A long-standing conjecture of Lapidus states that under certain conditions, self-similar fractal sets fail to be Minkowski measurable if and only if they are of lattice type. It was shown by Falconer and Lapidus (working independently but both using renewal theory) that nonlattice self-similar subsets of \({\mathbb {R}}\) are Minkowski measurable, and the converse was shown by Lapidus and v. Frankenhuijsen a few years later, using complex dimensions. Around that time, Gatzouras used renewal theory to show that nonlattice self-similar subsets of \({\mathbb {R}}^d\) that satisfy the open set condition are Minkowski measurable for \(d \ge 1\). Since then, much effort has been made to prove the converse. In this paper, we prove a partial converse by means of renewal theory. Our proof allows us to recover several previous results in this regard, but is much shorter and extends to a more general setting; several technical conditions appearing in previous work have been removed.     

Asymptotics of random sums of negatively dependent random variables in the presence of dominatedly varying tails

Let X ₁ , X ₂ , . . . be a sequence of negatively dependent and identically distributed random variables, and let N be a counting random variable independent of X _i ’s. In this paper, we study the asymptotics for the tail probability of the random sum S_N = ?_{k = 1}^N X_k {S_N} = \sum\nolimits_{k = 1}^N {{X_k}} in the presence of heavy tails. We consider the following three cases: (i) P(N > x) = o(P(X ₁  > x)), and the distribution function (d.f.) of X ₁ is dominatedly varying; (ii) P(X ₁  > x) = o(P(N > x)), and the d.f. of N is dominatedly varying; (iii) the tails of X ₁ and N are asymptotically comparable and dominatedly varying.     

Budget-constrained maintenance in the building industry

Given records of building component renewals which occurredduring the year and also theage profile of the building stock,the theory of alternating renewal processes in discrete timeis used to deduce the underlying probability mass functionsof the natural lifespan of building components and of delaydue to limited resource. In an investigation into the economicsof renewal delay, the expected number of component renewalsin each year of the life of a building is deduced from the convolutionsof the probability mass function of natural lifespan: theseexpected numbers depend on the chosen delay. The nett presentvalue of the expected cost associated with each specified delayis calculated, so that a judgement can be made as to the optimumdelay (if any). These calculations are conveniently performedusing a spreadsheet macro.