Date of this Version
Journal für die Reine und Angewandte Mathematik
Hammersley  showed that if X1, X2, . . . is a sequence of independent identically distributed random variables whose common distribution is continuous, and if ln+(ln-) denotes the length of the longest increasing (decreasing) subsequence of X1, X2, . . ., Xn, then there is a constant c such that ln-⁄n½→ c and ln+⁄n½→ c in probability, as n → ∞. Kesten  showed that in fact there is almost sure convergence. Logan and Shepp  proved that c ≧ 2, and recently Versik and Kerov  have announced that c = 2.
The original and published work is available at: https://www.degruyter.com/view/j/crll.1979.issue-306/crll.1979.306.49/crll.1979.306.49.xml
Boyd, D. W., & Steele, J. M. (1979). Monotone Subsequences in the Sequence of Fractional Parts of Multiples of an Irrational. Journal für die Reine und Angewandte Mathematik, 1979 (306), 49-59. http://dx.doi.org/10.1515/crll.1979.306.49
Date Posted: 27 November 2017
This document has been peer reviewed.