Comparisons of optimal stopping values and prophet inequalities for negatively dependent random variables (Q1099879)

Consider a finite sequence of random variables \(Z_ 1,...,Z_ n\), such that E \(| Z_ i| <\infty\), and let T be the set of all stopping rules t over \((Z_ 1,...,Z_ n)\). The set of generalized stopping rules (for which the condition \(P(t\leq n)=1\) is dropped) is also considered. The optimal value is defined as \(V(Z)=\sup \{E Z_ t:\) \(t\in T\}\). Random variables \(Y_ 1,Y_ 2,..\). are considered, which satisfy the weak negative dependence condition: \[ (*)\quad P(Y_ i<a_ 1| \quad Y_ 1<a_ 1,...,y_{i-1}<a_{i-1})\leq P(Y_ i<a_ i). \] The two principal results of the paper are the following: 1) Consider \(Y_ 1,...,Y_ n\) satisfying condition (*), and let \(X_ 1,...,X_ n\) be independent random variables such that \(Y_ i\) and \(X_ i\) have the same marginal distributions \((i=1,...,n)\); then V(X)\(\leq V(Y)\). This result extends to generalized stopping rules, and to infinite sequences. 2) \textit{U. Krengel} and \textit{L. Sucheston} [in \textit{J. Kuelbs} (ed.), Probability on Banach spaces (1978; Zbl 0394.60002), pp. 197-266] show that for any sequence \(X_ 1,...,X_ n\) of nonnegative independent random variables, the prophet inequality \(E\{x_ 1\vee...\vee X_ n\}\leq 2V(X)\) holds, and that 2 is the smallest constant for which this is true. In this paper, this result is extended to nonnegative random variables satisfying condition (*). This result also extends to infinite sequences.