Higher-order Expansions Of Dynamic Bivariate Normal Copula

Let {(Xni,Yni),1 ≤ i ≤ n,n ≥ 1} be an independent triangular arrays,and the distribution of this triangular arrays follows the normal copula with correla-tion coefficient ρni= 1-m(i/n)/log n with an unknown smooth function m(s).Limit distribution of Mn =(n(max1≤i≤nF1(Xni)-1),(max1≤i≤nF2(Yni)-1)),the maxima of dynamic bivariate normal copula,has been studied.In this the-sis,higher-order expansions of distributions and densities of Mn are considered under the following three cases:m(s)is a continuous positive function on[0,1],limn→∞ min1≤i≤n m(i/n)= ∞= and lim n→∞ max1≤i≤nm(i/n)= 0,respectively.The main results show that the convergence rates of distributions and densities are the same order.The thesis is organized as the following two parts.In the first part of this thesis,the second-order distributional expansions of Mn are firstly established when m(s)is a continuous positive function on[0,1].Secondly,the second-order distributional expansions of Mn are derived by imposing some constraints as limn→∞ min1≤i≤n m(i/n)= ∞＝ and limn→∞ max1≤i≤nm(i/n)= 0,respectively.Finally,the convergence rates of distributions of extremes of dynamic bivariate normal copula are derived.For the second part,the asymptotics of densities of extremes of dynamic bi-variate normal copula are studied.Limits of densities of Mn are firstly considered with m(s)in different cases.Then the second-order expansions of densities of Mn are derived under some additional conditions.Finally,the rates of convergence of densities of extremes of dynamic bivariate normal copula are obtained.