Optimal Sequential Selection of a Monotone Sequence From a Random Sample
Stephen M. Samuels, John M. Steele · The Annals of Probability · 1981
The length of the longest monotone increasing subsequence of a random sample of size $n$ is known to have expected value asymptotic to $2n^{1/2}$. We prove that it is possible to make sequential choices which give an increasing subsequence of expected length asymptotic to $(2n)^{1/2}$. Moreover, this rate of increase is proved to be asymptotically best possible.