Limit Theorems On Gaussian Vector Sequences

This thesis focuses on the study of weak convergence and almost sure convergence of Gaussian vector sequences. The first part of this paper, the joint limit distribution of maximum and minimum of a standard stationary Gaussian sequence with random index was analyzed, and the almost sure central limit theorem of maximum and minimum of a standard stationary Gaussian vector sequence was obtained. The main results are as follows:Theorem A Suppose that {X_n} be a standard stationary Gaussian sequence satisfying(2.3). Also let N(n) be a positive integer valued random variable satisfying(2.2).Assumed that there are some numerical sequences {u_n},{v_n} satisfying (2.4). ThenTheorem B Suppose that {X_n} be a standard stationary Gaussian sequence of d-dimensional random vectors satisfying (2.13) and (2.14). (1). Assumed that there are some numerical sequences {u_ns}, {v_ns}, 1≤s≤d satisfying (2.15), thenIn the second part, going on the discussion of the almost sure central limit theorem of the partial sum and maximum of a standard stationary Gaussian vector sequence, we considered the limit distribution of point process of exceedances of a kind of standard stationary Gaussian vector sequence and its asymptotical independence with the partial sum of this Gaussian vector sequence, which are:Theorem C Suppose that {X_n} be a standard stationary Gaussian sequence of d-dimensional random vectors satisfying (3.4), (3.5) and (3.6).(1) Assumed that there are some numerical sequences {u_ns}, 1≤s≤d satisfying (3.7). thenTheorem D Let {X_n} be a standard stationary Gaussian sequence of d-dimensional random vectors satisfying (3.14). (3.15) and (3.16). Assumed that there are some numerical sequences {u_ns} satisfying (3.17), N_n is the point process of exceedances of levels {u_ns} by {X_n}. Then N_n(?)N as n→∞, where Nis a Poisson vector point process on (0.1] with intensity sum from s=1 to d m(B_s)e^-x_s and independent components. A byproduct is the asymptotically independence betweenN_n and (S_n)/(σ_n).