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Almost sure convergence for the maximum and the sum of nonstationary Gaussian sequences. (Almost sure convergence for the maximum and the sum of nonstationary guassian sequences.) (English) Zbl 1195.60047

Summary: Let (\(X_{n}, n\geq 1\)) be a standardized nonstationary Gaussian sequence. Let \(M_{n}= \max\{X_{k},1\leq k\leq n\}\) denote the partial maximum and \(S_{n}=\sum _{k=1}^{n}X_{k}\) for the partial sum with \(\sigma _{n}=\) (Var \(S_{n})^{1/2}\). In this paper, the almost sure convergence of \((M_{n}, S_{n}/\sigma _{n})\) is derived under some mild conditions.

MSC:

60F15 Strong limit theorems
60G70 Extreme value theory; extremal stochastic processes
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