Closure Property Of Random Sum And Its Maximum Of Random Variables Based On Precise Large Deviation Principles

The research on the heavy-tailed risk model in financial insurance risk management has been hot,This is because the bankruptcy of an insurance company is not often caused by a large amount of small claims,but the fact that the claims of an extreme event far exceed the actual claims ability of the insurance company.The extreme events refer to natural disasters or man-made major disasters on a global scale.The claims of such an event can account for more than 80 of the insurance company's compensation,and even lead to bankruptcy.Under extreme events the heavy-tailed distribution is very effective in portraying the distribution of claims.It can provide warning data in a timely manner,so that insurance companies can prevent extreme risks in advance,and it is meaningful to achieve stable operation and sustainable development.Due to the importance of heavy-tailed distribution in the risk model,and many problems in this field can be attributed to large deviation problems.In recent years,it is also a hot topic to study the random sum and its maximum value of heavy-tailed random variable sequences in the field of actuarial science.This article also explores around it.This paper mainly studies two important heavy-tailed distributions C and D classes,including three parts:The first part is mainly under the condition that the random variables sequence {X1,X2,...} is independent,it improved the results of the Theo-rem 6 of Kizinevic et al.(2016)[19]and the Theorem 5 of Andrulyte et al.(2017)[2]by adding some other mild tail restrictions,and applying the precise large deviation principles for the sums of the random variables sequence with the classes of C,obtain closure property of random sum and its maximum of random variables;The second part is also independent of the random variables sequence(X1,X2,...},it improved the results of the Theorem 2.1 of Danilenko and Siaulys(2016)[12]and the Theorem 2 of Leipus and Siaulys(2017)|22]by add some other mild tail restrictions on ? too,applying the precise large deviation principles derived S? and S(?)satisfied the closure of the class D;The third part is about the closure property of weighted random vec-tores sum and its maximum.Let {(?1,X1),(?2,X2),...·} be mutually ran-dom vectors,whereas some dependence structure exists each pair {(?k,Xk),k 1,2,...}.By using some lemma,we can get 0kXk are still belongs to the same distribution,but because{(?k,Xk),k=1,2,...} are independent of each other,so to return to the first and second conditions,we get the weighted random sum and its maximum in C and D classes.The innovation of this paper is applying the precise large deviation as a tool to prove the closure of C and D classes,meanwhile relaxation of the condition requirement of ? in the theorem,which is an important supplement of the above four literature results.