Let X1, X2, ⋯, Xn, ⋯ be a sequence of i.i.d. random variables uniformly distributed on [0; 1], and denote by Ln the length of the longest increasing subsequences of X1, X2, ⋯, Xn. Consider the poissonized version Hn based on Hammersley’s representation in the 2-dimensional space. A law of the iterated logarithm for Hn is established using the well-known subsequence method and Borel-Cantelli lemma. The key technical ingredients in the argument include superadditivity, increment independence and precise tail estimates for the Hn’s. The work was motivated by recent works due to Ledoux (J. Theoret. Probab.31, (2018)). It remains open to establish an analog for the Ln itself.

中文翻译：

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最长递增子序列长度的 LIL

设 X1, X2, ⋯, Xn, ⋯ 是在 [0; 上均匀分布的 iid 随机变量序列；1]，并用 Ln 表示 X1、X2、⋯、Xn 的最长递增子序列的长度。考虑基于 Hammersley 在二维空间中的表示的泊松化版本 Hn。使用众所周知的子序列方法和 Borel-Cantelli 引理建立了 Hn 的迭代对数定律。论证中的关键技术成分包括超可加性、增量独立性和 Hn 的精确尾部估计。这项工作受到 Ledoux (J. Theoret. Probab.31, (2018)) 近期作品的启发。为 Ln 本身建立一个类似物仍然是开放的。