Asymptotics and renewal approximation in the online selection of increasing subsequence

On sequential selection and a first passage problem for the Poisson process

“…Bruss and Delbaen [7] showed that L(t) is AN( √ 2t, 1 3 √ 2t) (AN abbreviates 'asymptotically normal'), see also [13]. We argue that the same is true for N(t).…”

“…see [5,7,13]. In [13] we proved that every policy having ψ(z) ∼ 2/z is within O(1) from the optimality, that is achieves the asymptotics (2).…”

“…see [5,7,13]. In [13] we proved that every policy having ψ(z) ∼ 2/z is within O(1) from the optimality, that is achieves the asymptotics (2). The general Markovian policy differs from ( 4) and ( 5) in that ψ is replaced by the general function of τ, t and x.…”

“…In [13] we called such policies self-similar because the performance from each stage on only depends on the mean number of future acceptable arrivals. Thus defined, I is a jump Markov process with transition mechanism determined by ψ.…”

“…See Romik's book [18] for a nice exposition. The leading term of ( 2) is due to Samuels and Steele [19] who were first to study the online problem, later Bruss and Delbaen [7] identified the logarithmic order of the second term and the full expansion has appeared recently in [13].…”