Precise Large Deviations For The Difference Of Two Sums Of Dependent Random Variables With Heavy Tails

In actuarial mathematics heavy-tailed distribution is commonly used to describe the nature of extreme events,and precise large deviation of the sums of random variables with heavy tails,as a highlighted problem,is concerned by many scholars.Such as in the case that two insur-ance forms are promoted within an insurance company and these two forms are {X1i,i? 1} and{X2i,i ?1},when x??{(?j=1n1X1j-?j=1n2X2j)x},it represents that one type of insurance claim is much larger than the other and the former is likely to make the insurance company bankrupt and therefore the insurance company for the former should be paid more attention.So far,there are few researches about asymptotic behavior of ?k=1n1X1j-?j=1n2X2j.Based on the existing research results,the problem of precise large deviation of the difference of two types of insurance contracts is studied,and then the problem of precise large deviation of random sum of UEND and ?-mixing random variables is investigated,precise large deviations for claim risk model and claim surplus risk model are considered at last.In this paper,the main research con-tents include the following several aspects:Firstly,let {X1j,j? 1} be a sequence of NA and nonnegative and identically distributed random variables with common distribution function F1?C,{X2j,j ?1} be a sequence of non-negative and independent and identically distributed random variables.For i =1,2,ni(t)is a positive integer-valued function which satisfies ni(t)?? as t??.We study the tail probabil-ity for ?j=1n1(t)X1j-?j=1n2(t)X2j to obtain the asymptotical conclusion of precise large deviation for?j=1n1(t)X1j-?j=1n2(t)X2j under some conditions,and extend the existing corresponding conclusion under the conditions of independent and identically distributed random variables.Secondly,for i = 1,2,let {Xij,j ? 1} be a sequence of END and nonnegative and iden-tically distributed random variables with common distribution function Fi and finite mean ?i.We study precise large deviations for ?j=1n1(t)X1j-?j=1n2(t)X2j and ?j=1N1(t)X1j-?j=1N2(t)X2j under the some advanced given conditions which are F1?C and arbitrary distribution F2 combined with appropriate series of other conditions,where{Ni(t),t ?0}i=1,2 and {Xij,j?1}1,2 are mutually independent,and {Ni(t),t ?0}i?1,2 are non-negative integer-valued counting processes.Thirdly,we study large deviations for ?j=1n1(t)X1j-?j=1n2(t)X2j and ?j=1N1(t)X1j-?j=1N2(t)X2j,where{X1j,j ? 1} is a sequence of WUOD and nonnegative and non-identically distributed random variables,and {X2j,j?1} is a sequence of nonnegative and independent and identically distribut-ed random variables.For i= 1,2,ni(t)is a positive integer-valued function,and {Ni(t),t?0}i=1,2 with ENi(t)=?i(t)are two counting processes.We obtain the asymptotical conclusions of precise large deviation for ?j=1n1(t)X1j-?j=1n2(t)X2j and ?j=1N1(t)X1j-?j=1N2(t)X2j when some other as-sumptions are satisfied,and extend the existed corresponding conclusions.Fourthly,we study the issue of the tail probability of random sum of UEND and ?-mixing random variables,and obtain the asymptotical conclusion of precise large deviation of random sum of non-identically distributed and UEND and 0-mixing random variables by the similar method which is used to derive the asymptotical conclusion of precise large deviation of ran-dom sum of dependent and non-identically distributed random variables.This asymptotical conclusion are extended from independent and non-identically distributed random variables to dependent and non-identically distributed random variables.Fifthly,{Xk,k?1} is a sequence of non-identically distributed and END and ?-mixing random variables with common distribution function which belongs to D,and denotes a claim process;{Yj,j?1} is a sequence of nonnegative and identically distributed and END random variables,and denotes a premium process.We consider the claim risk model constructed by{Xk,k?1} and study the asymptotic problem of the tail probability of non-random sum and random sum in this claim risk model in order to obtain the asymptotical conclusions of precise large deviation of non-random sum and random sum in this model by the similar method which is used to derive the asymptotic formula of precise large deviations of non-random sum and ran-dom sum of dependent and non-identically distributed random variables.And then,we consider the claim surplus risk model constructed by the claim process(Xk,k?1} and the premium pro-cess {Yj,j?1} to obtain the asymptotic formula of precise large deviation in claim surplus risk model.