Asymptotic theory of selection by relative rank with high cost

Selection from among n objects by relative rank with no recall - the ‘secretary problem’ - in the asymptotic case when is considered, assuming that k, the cost ratio, becomes very large and is kept to be a finite constant. It is known that in the ‘low cost’ case where K=kn is kept to be a non-negative finite constant, the expected number of observations comes out to be O(n) and the expected absolute rank of the selected object comes out to be O(1), and that in the ‘medium cost’ case where kis kept to be a positive finite constant, both expected values come out to be . Here the former comes out to be O(1) and the latter O(n). teh graph of total ‘loss’ vs. K looks like a continuous broken line, while those of ‘expected cost of observations’ and ‘expected absolute rank’ have jumps at many points. As k roaches 0, the curves approach smooth ones corresponding to the ‘medium cost’ case.