Assaf, David; Samuel-Cahn, Ester Prophet inequalities for optimal stopping rules with probabilistic recall. (English) Zbl 1006.60036 Bernoulli 8, No. 1, 39-52 (2002). The paper deals with an extension of the prophet problems when statistician is able to solicit an observation not chosen in the past when it has appeared. A sequence of random variables \(X_1,X_2,\ldots,X_n\) are observed sequentially and the probability vector \({\mathbf p}=(p_1,p_2,\ldots,p_{n-1})\), where \(1\geq p_1\geq p_2\geq\ldots\geq p_{n-1}\geq 0\), is given. Two models of availability are considered. (i) If an item \(X_i\) is not available at time \(k>i\), it will remain unavailable beyond \(k\). (ii) The items are available by probabilistic recall using the vector \(({\mathbf p})\). Three modes of stopping are taken into account. Only one object may be chosen. Let \(V_{\mathbf{p}}(X_1,\ldots,X_n)\) be the optimal value to statistician. It is shown that for all non-trivial, non-negative \(X_i\) and all \(n\geq 2\) we have \[ \mathbf{E}\max(X_1,\ldots,X_n)<(2-p_{n-1})V_{\mathbf{p}}(X_1,\ldots,X_n) \] and \(2-p_{n-1}\) is the best constant [see U. Krengel and L. Sucheston, in: Problems on Banach spaces. Adv. Probab. Relat. Top. 4, 197-266 (1978) and T. P. Hill and R. P. Kertz, Z. Wahrscheinlichkeitstheorie Verw. Geb. 56, 283-285 (1981; Zbl 0443.60039) for the classical ratio prophet inequality]. It is proved also that for bounded independent random variables such that \(a\leq X_i\leq b\), \(i=1,2,\ldots,n\), we have \[ \mathbf{E}\max(X_1,\ldots,X_n)-V_{\mathbf{p}}(X_1,\ldots,X_n)\leq \left[\frac{1-\sqrt{1-p_{n-1}}}{p_{n-1}}\right]^2(1-p_{n-1})(b-a). \] It is a generalization of the result by T. P. Hill and R. P. Kertz (loc. cit.). Reviewer: K.Szajowski (Wrocław) Cited in 1 Document MSC: 60G40 Stopping times; optimal stopping problems; gambling theory 62L15 Optimal stopping in statistics Keywords:optimal stopping; prophet regions; probabilistic recall; iid problem; backward solicitation; balayage technique Citations:Zbl 0443.60039 PDFBibTeX XMLCite \textit{D. Assaf} and \textit{E. Samuel-Cahn}, Bernoulli 8, No. 1, 39--52 (2002; Zbl 1006.60036)