Consistency Of CVaR Optimal Estimator Under Optimal Moment Conditions

As we all know,risk is uncertain and Conditional Value-at-Risk(CVaR)is one of commonly used risk measures,and probability theory is the branch of the study of the possibility of random events in mathematics.We prove the consistency and convergence rate for the optimal estimation of CVaR under the optimal moment condition,which is based on the basic theory of modern probability theory.Suppose that Z is a collectivity and ?Z,Zn,n?1} is a simple random sample from Z.Defining the true values of CVaR and the optimal estimator are where[x]+=max{0,x},x?R,?n(t)=n-1?n i=1[Zi-t]+.Through studying these convergence properties of the estimator (?)n,the paper proves the following results.If the first moment of the population exists,the paper obtains the strong consistency for the estimator (?)n.That is (?)n??*a.s.if E|Z|<+?.And then we study the mean convergence and the convergence rate of the weak consistency about the estimator,which proves the following conclusions.First,it is that let { Z,Zn,n?1 } be a sequence of independent and identically distributed random variables with xpP(|Z|>x)?0} for p>1,then P(|(?)n-?*|>?)=o(n-(p-1));Second,it is that let {Z,Zn,n?1? be a sequence of independent and identically distributed random variables with E|Z|p<? forp?1,thenThird,it is that let {Z,Zn,n?l } be a sequence of independent and identically distributed random variables with E|Z|p<·? for1p<2,then E|(?)n-?*|p?0.At last,the paper verifies the above conclusion through the numerical simulation.And the paper makes an empirical analysis of Shenzhen A shares by using these conclusions.