Cauchy-Schwarz inequalities from the viewpoint of lower and upper previsions

In recent papers we investigated how some classical probability inequalities still obtain, in a modified form, when bounded random numbers, also termed gambles, are evaluated by means of upper/lower (imprecise) previsions satisfying various consistency criteria. This kind of analysis is developed in this paper with reference to the Cauchy-Schwarz (CS) inequality. We show that two among the potential extensions of the CS inequality obtain without restrictions when the upper/lower previsions are coherent. We then consider weaker consistency requirements than coherence. When the imprecise previsions avoid sure loss or are centered convex, these two inequalities may fail to hold; nevertheless, we give sufficient conditions for them to apply. With 2-coherent upper/lower previsions, we provide several sufficient conditions for the two inequalities to hold, limiting the instances where they fail to rather peculiar cases. We also show that it is possible to obtain, in certain instances, information about the correlation of two gambles without requiring coherence.