Asymptotic Distributions Of Maxima Of Complete And Incomplete From Stationary Gaussian Sequences

The asymptotic behavior of distributions of maxima for Gaussian sequences depend on the convergence rates of its covariance, and it is known that there are three limiting distributions for the Gaussian sequences. This thesis focuses on the joint limiting distribution of the maxima of complete and incomplete samples of stationary Gaussian sequences.In the first part of the thesis, we consider the joint limit distributions of ex-tremes of complete and incomplete samples of two classes of Gaussian vector se-quences. Let {Xn = (Xn1,Xn2,…,Xnd), n≥1} be a sequence of d-dimensional stationary Gaussian vectors with EXni = 0. Var (Xni) = 1 and Cov (Xli,Xkj) rij(|l-k|). Let Mn = (Mn1,Mn2,…,Mnd) denote the partial maxima of {Xk} i.e. Mni=max{X1i, X2i,…, Xni}. Suppose that some of the variables of Xk can be observed. Letεki denote the indicator of the event that Xki is observed. Let Sni =ε1i +ε2i +…+εni be the number of observed random variables from the set {X1i, X2i,…, Xni}. Let Mn denote the partial maxima of the observed variables. Assume thatεki, 1≤k≤n. 1≤i≤d are independent random variables, arid also independent of {Xk}, and the sequence (Sni) satisfies Sni/n(?)pi∈(0, 1] as n→∞for 1≤i≤d. Then, we get the joint limiting distributions of Mn and Mn as rij(n) log n→0 and rij(n)log n→ρij∈(0.∞) rcspectivcly.For the second part, we study the joint limiting distribution of maxima of complete and incomplete samples of strongly dependent Gaussian sequence. Let {Xn} be a stationary Gaussian sequcnce with E Xn = 0, Var (Xn) = 1. Let Mn = max1≤k≤Xk and rn = E X1Xn+1 denote the partial maximum and the correlation respectively. Suppose that some of the variables of {X1,…, Xn} can be observed. Letεk denote an indicator variable and {εk = 1} mean the event that the random variable Xk is observed, and Sn denotes the number ofobserved random variables from the set {X1,…,Xn}. Let Mn denote the partial maxima of the observed variables. Assume thatεk, 1≤k≤n, are independent: and also independent of {Xk}. The sequence (Sn) satisfies Sn/n(?)p∈(0, 1]. Assume further that rn satisfies following two conditions: rn is convex with rn = o(1); and (rn log n)-1 is monotone with (rn log n)-1 = o(1) as n→∞. We get the joint limiting distribution of Mn and Mn for the strongly dependent Gaussian sequence with the assumed conditions.