Abstract
Let {Xk,k⩾1} be a stationary Gaussian sequence with partial maximum Mn=max {Xk,1⩽k⩽n} and sample mean X̅n=∑k=1nXk/n. Suppose that some of the random variables X1,X2,… can be observed and the others not. Denote by M̃n the maximum of the observed random variables from the set {X1,X2,…,Xn}. Under some mild conditions, we prove the joint limiting distribution and the almost sure limit theorem for (M̃n−X̅n,Mn−X̅n).
Highlights
Let {Xk, k 1} be a standardized stationary Gaussian sequence
We prove the joint limiting distribution and the almost sure limit theorem for (Mn − Xn, Mn − Xn)
The limiting distribution of Mn for weakly dependent stationary Gaussian sequences has been studied by Berman
Summary
Let {Xk, k 1} be a standardized stationary Gaussian sequence. Let rn = E X1Xn+1 and Mn = max{Xk, 1 k n}. The limiting distribution of Mn for weakly dependent stationary Gaussian sequences has been studied by Berman [2], i.e., lim n→∞. For the limiting distribution of the partial maxima of strongly dependent stationary Gaussian sequences, see Lin [10] and Mittal and Ylvisaker [14] for the case of rn log n → γ ∈ (0, ∞) and McCormick and Mittal [13] for the case rn log n → ∞ with some additional conditions. We are interested in the joint limiting distribution and the almost sure limit theorem (ASLT) of maxima centered at sample mean for complete and incomplete samples from stationary Gaussian sequences. For weakly dependent stationary Gaussian sequences, Csaki and Gonchigdanzan [6] showed that lim n→∞.
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