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Comparisons of optimal stopping values and prophet inequalities for negatively dependent random variables. (English) Zbl 0639.60052
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) U. Krengel and L. Sucheston [in 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.
Reviewer: J.Gianini-Pettitt

##### MSC:
 60G40 Stopping times; optimal stopping problems; gambling theory 60E15 Inequalities; stochastic orderings 60K30 Applications of queueing theory (congestion, allocation, storage, traffic, etc.)
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