Abstract
Let X 1,X 2, ⋯ be a sequence of independent and identically distributed random variables and M n = max {X 1,X 2, ⋯ , X n }. Suppose that some of the random variables X 1,X 2, ⋯ , X n can be observed and denote by \(\widetilde{M}_{n}\) the maximum of the observed random variables from the set {X 1,X 2, ⋯ , X n }. The limiting distribution of random vector \((M_{n}-\widetilde{M}_{n}, \widetilde{M}_{n})\) is derived. The result is also extended to the case of stationary Gaussian sequences. In the end, the almost sure limit theorem on \(M_{n}-\widetilde{M}_{n}\) for a sequence of independent and identically distributed random variables is proved.
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Research was supported by National Science Foundation of China (No. 11071182).
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Tan, Z., Wang, Y. Some asymptotic results on extremes of incomplete samples. Extremes 15, 319–332 (2012). https://doi.org/10.1007/s10687-011-0140-z
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DOI: https://doi.org/10.1007/s10687-011-0140-z