随机结构中的极限定理
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摘要
本文主要分为三个部分,我们分别研究在金融保险风险模型,再保险模型,和随机图模型中的一些极限定理.
在金融保险行业中,两随机变量X和Y的乘积Z=XY的尾分布行为研究一直是一项基础课题,并得到了大量的应用.然而迄今为止,几乎所有的结果都是建立在随机变量X和Y间相互独立的假设前提下的,事实证明这个假设是是非常不现实的.本文在假设两随机变量间服从一定的相依结构下,考察了它们乘积的尾分布与独立情形下乘积尾分布的渐近关系,我们感兴趣的是如何抓住随机变量间的相依结构对其乘积尾分布行为的沖击因子.特别地,对相依结构服从广义FGM分布时,我们得到随机变量X和Y乘积Z的尾概率的明确的渐近公式,与独立情形相比较,我们的结果包含了一个透明的因子来表示X和Y间相依结构对其乘积的冲击.更进一步,我们深入研究了在此相依结构下保险模型中的破产概率问题。
另外,我们考察了在大额再保险模型LCR和ECOMOR中,再保险额L_ι(t)和E_ι(t)的尾分布行为.在ι和t固定的条件下,我们得到了L_ι(t)和E_ι(t)的尾概率准确的渐近估计.我们的结果显示,当单个索赔额分布F为指数分布时,上述尾概率均与gamma分布尾概率与某因子的乘积渐近等价.而当F具有自卷积尾平衡分布时,上述尾概率均与F的尾分布和某因子的乘积渐近等价.其中,被乘上的因子都是完全透明的.
最后,我们考察了在随机图结构中的某些极限定理,如,各种单边区间树的最大间隔的极限性质,以及Buckley-Osthus无标度图的度数序列中的极限性质,另外,我们还考察了它的最大度数满足的极限定理.
This dissertation consists of three parts.We establish some limit laws for the insurance, reinsurance,and random graphs models,respectively.
In insurance and finance,the product Z=XY of two random variables X and Y is one of basic elements in stochastic modelling.There were many works about the tail behavior of the product under the assumption that the random variables are independent.However,this assumption is far too unrealistic for most applied problems.Therefore,it is more interesting and practical to study the case that X and Y are dependent.We assume that X and Y follow a generalized FGM distribution.We are interested in the question how to capture the impact of the dependence of X and Y in this model on the tail behavior of their product Z.We shall derive an explicit asymptotic formula for the tail probability of Z.In comparison to the asymptotic formula for the independent case,ours contains an extra factor representing the impact of the dependence of X and Y.
We shall also investigate the asymptotic behavior of the tail probabilities of L_ι(t) and E_ι(t),which are reinsurance amounts in large claims reinsurance models LCR and ECOMOR, respectively.We will establish precise asymptotic estimates for the tail probabilities of L_ι(t) and E_ι(t),withιand t fixed.Our results show that,when F,the the distribution of claim size,is an exponential distribution,these tail probabilities are both asymptotic to a multiple of the tail of a gamma distribution with suitable parameters,while when F has a convolution-equivalent tail,they are both asymptotic to a multiple of the tail of F.The prefactors involved are completely explicit and transparent.
At last,we will obtain some limit theorems in random graphs,for example,the maximal gap in division of one-sided interval trees,the degree sequences of Buckley-Osthus scale-free random graphs,and so on.
在金融保险行业中,两随机变量X和Y的乘积Z=XY的尾分布行为研究一直是一项基础课题,并得到了大量的应用.然而迄今为止,几乎所有的结果都是建立在随机变量X和Y间相互独立的假设前提下的,事实证明这个假设是是非常不现实的.本文在假设两随机变量间服从一定的相依结构下,考察了它们乘积的尾分布与独立情形下乘积尾分布的渐近关系,我们感兴趣的是如何抓住随机变量间的相依结构对其乘积尾分布行为的沖击因子.特别地,对相依结构服从广义FGM分布时,我们得到随机变量X和Y乘积Z的尾概率的明确的渐近公式,与独立情形相比较,我们的结果包含了一个透明的因子来表示X和Y间相依结构对其乘积的冲击.更进一步,我们深入研究了在此相依结构下保险模型中的破产概率问题。
另外,我们考察了在大额再保险模型LCR和ECOMOR中,再保险额L_ι(t)和E_ι(t)的尾分布行为.在ι和t固定的条件下,我们得到了L_ι(t)和E_ι(t)的尾概率准确的渐近估计.我们的结果显示,当单个索赔额分布F为指数分布时,上述尾概率均与gamma分布尾概率与某因子的乘积渐近等价.而当F具有自卷积尾平衡分布时,上述尾概率均与F的尾分布和某因子的乘积渐近等价.其中,被乘上的因子都是完全透明的.
最后,我们考察了在随机图结构中的某些极限定理,如,各种单边区间树的最大间隔的极限性质,以及Buckley-Osthus无标度图的度数序列中的极限性质,另外,我们还考察了它的最大度数满足的极限定理.
This dissertation consists of three parts.We establish some limit laws for the insurance, reinsurance,and random graphs models,respectively.
In insurance and finance,the product Z=XY of two random variables X and Y is one of basic elements in stochastic modelling.There were many works about the tail behavior of the product under the assumption that the random variables are independent.However,this assumption is far too unrealistic for most applied problems.Therefore,it is more interesting and practical to study the case that X and Y are dependent.We assume that X and Y follow a generalized FGM distribution.We are interested in the question how to capture the impact of the dependence of X and Y in this model on the tail behavior of their product Z.We shall derive an explicit asymptotic formula for the tail probability of Z.In comparison to the asymptotic formula for the independent case,ours contains an extra factor representing the impact of the dependence of X and Y.
We shall also investigate the asymptotic behavior of the tail probabilities of L_ι(t) and E_ι(t),which are reinsurance amounts in large claims reinsurance models LCR and ECOMOR, respectively.We will establish precise asymptotic estimates for the tail probabilities of L_ι(t) and E_ι(t),withιand t fixed.Our results show that,when F,the the distribution of claim size,is an exponential distribution,these tail probabilities are both asymptotic to a multiple of the tail of a gamma distribution with suitable parameters,while when F has a convolution-equivalent tail,they are both asymptotic to a multiple of the tail of F.The prefactors involved are completely explicit and transparent.
At last,we will obtain some limit theorems in random graphs,for example,the maximal gap in division of one-sided interval trees,the degree sequences of Buckley-Osthus scale-free random graphs,and so on.
引文
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[2]Ammeter,H.(1964).The rating of largest claim reinsurance covers.Quart.Algem.Reinsur.Comp.Jubilee.,no.2,5-17.
[3]Asimit,A.V.and Jones,B.L.(2008).Asymptotic tail probabilities for large claims reinsurance of a portfolio of dependent risks.Astin Bull.,38,no.1,147-159.
[4]Asmussen,S.and Rojas-Nandayapay,L.(2008).Sums of dependent lognormal random variables:asymptotics and simulation.Statist.Probab.Lett.,to appear.
[5]Bairamov,I.,Kotz,S.,and Bekci,M.(2001).New generalized Farlie-Gumbel-Morgenstern distributions and concomitants of order statistics.J.Appl.Stat.,28,no.5,521-536.
[6]Barab(?)si,A.L.and Albert,R.(1999).Emergence of scaling in random networks.Science,286,509-512.
[7]Barbe,Ph.and McCormick,W.P.(2008).Asymptotic expansions for infinite weighted convolutions of rapidly varying subexponential distributions.Probab.Theory.Relat.Fields,to appear.
[8]Beirlant,J.and Teugels,J.L.(1992).Limit distributions for compounded sums of extreme order statistics.J.Appl.Probab.,29,no.3,557-574.
[9]Bingham,N.H.,Goldie,C.M.and Teugels,J.L.(1987).Regular Variation.Cambridge University Press,Cambridge.
[10]Bollob(?)s,B.(1998).Modern Graph Theory.Springer-Verlag,New York.
[11]Bollob(?)s,B.and Riordan,O.(2003).Mathematical results on scale-free random graphs.In S.Bornholdt and H.G.Schuster,editors,Handbook of Graphs and Networks,Wiley,1-34.
[12]Bollob(?)s,B.and Riodan,O.(2004).The diametre of a scale-free random graphs.Combinatorica,24,no.1,5-34.
[13]Borovkov,A.A.(1984).Asymptotic Methods in Queueing Theory.Springer,New York.
[14]Borovkov,A.A.(1976).Stochastic Processes in Queueing.Springer,New York.
[15]Bollob(?)s,B.,Riordan,O.,Spencer,J.,and Tusnady,G.(2001).The degree sequence of a scale-free random graph process.Random Structuress Algorithms,18,279-290.
[16]Breiman,L.(1965).On some limit theorems similar to the arc-sin law.Theory Prob.Appl.,10,323-331.
[17]Buckley,P.G.and Osthus,D.(2004).Popularity based random graph models leading to a scalefree degree sequence.Discr.Maths.,282,53-68.
[18]Cai,J.(2002).Ruin probabilities with dependent rates of interest.J.Appl.Probab.,39,no.2,312-323.
[19]Chistyakov,V.P.(1964).A theorem on sums of independent positive random variables and its applications to branching random processes.Theory Probab.Appl.,9,640-648.
[20]Chover,J.,Ney,P.and Wainger,S.(1973a).Functions of probability measures.J.Analyse Math.,26,255-302.
[21]Chover,J.,Ney,P.and Wainger,S.(1973b).Degeneracy properties of subcritical branching processes.Ann.Probability,1,663-673.
[22]Cline,D.B.H.(1986).Convolution tails,product tails and domains of attraction.Probab.Theory Relat.Fields,72,no.4,529-557.
[23]Cline,D.B.H.(1989).Consistency for least squares regression estimators with infinite variance.J.Statist.Plann.Inference,23,163-179.
[24]Cline,D.B.H.and Samorodnitsky,G.(1994).Subexponentiality of the product of independent random variables.Stochastic Process.Appl.,49,no.1,75-98.
[25]Davis,R.A.and Resnick,S.I.(1985).Limit theory for moving averages of random variables with regularly varying tail probabilities.Ann.Probab.,13,179-195.
[26]Davis,R.A.and Resnick,S.I.(1986).Limit theory for the sample covariance and correlation function of moving averages.Ann.Statist.,14,533-558.
[27]Denisov,D.and Zwart,B.(2007).On a theorem of Breiman and a class of random difference equations.J.Appl.Probab.,44,no.4,1031-1046.
[28]Dorogovtsev,S.N.,Mendes,J.F.F.,and Samukhin,A.N.(2000).Structure of growing networks:Exact solution of the Barabasi-Albert Model.Phys.Rev.Lett,85,4633.
[29]Embrechts,P.(1983).A property of the generalized inverse Gaussian distribution with some applications.J.Appl.Probab.,20,no.3,537-544.
[30]Embrechts,P.and Goldie,C.M.(1980).On closure and factorization properties of subexponential and related distributions.J.Austral.Math.Soc.Set.A,29,no.2,243-256.
[31]Embrechts,P.,Kliippelberg,C.and Mikosch,T.(1997).Modelling Extremal Events for Insurance and Finance.Springer-Verlag,Berlin.
[32]Embrechts,P.;Veraverbeke,N.(1982).Estimates for the probability of ruin with special emphasis on the possibility of large claims.Insurance Math.Econom.,1,no.1,55-72.
[33]Farlie,D.J.G.(1960).The performance of some correlation coefficients for a general bivariate distribution.Biometrika,47,307-323.
[34]Fisher,N.I.(1997).Copulas.Encyclopedia of Statistical Sciences,Update Vol.1,159-163.
[35]Foss,S.and Korshunov,D.(2007).Lower limits and equivalences for convolution tails.Ann.Probab.,35,no.1,366-383.
[36]Goldie,C.M.(1991).Implicit renewal theory and tails of solutions of random equations.Ann.Appl.Probab.,1,no.1,126-166.
[37]Grey,D.R.(1994).Regtdar variation in the tail behaviour of solutions of random difference equations.Ann.Appl.Probab.,4,no.1,169-183.
[38]Grincevi(?)ius,A.K.(1975).One limit distribution for a random work on the line.Lithuanian Math.J.,15 580-589.
[39]Gumbel,E.J.(1960).Bivariate exponential distributions.J.Amer.Statist.Assoc.,55,698-707.
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