A difference prophet inequality for bounded i.i.d. variables, with cost for observations (Q1201170)

The class of the difference prophet inequalities for the sequence of random variables \(Y_ i=X_ i-ic\), \(i=1,2,\dots\), where \(X_ i\), \(i=1,2,\dots\), are i.i.d. random variables such that \(0\leq X_ i\leq 1\) is studied. The comparison with the strictly related results of \textit{M. Jones} [J. Multivariate Anal. 34, No. 2, 238-253 (1990; Zbl 0753.60042)] for the case when \(X_ i\), \(i=1,2,\dots\), are independent random variables is given. The review of the prophet problems in the optimal stopping can be found in the paper by \textit{T. P. Hill} and \textit{R. P. Kertz} [Strategies for sequential search and selection in real time, Proc. Conf., Amherst/MA (USA) 1990, Contemp. Math. 125, 191-207 (1992)].