LIL for the Length of the Longest Increasing Subsequences

Abstract

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.