  In many areas of engineering and sciences, decision rules and control
strategies are usually designed based on nominal values of relevant system
parameters. To ensure that a control strategy or decision rule will work
properly when the relevant parameters vary within certain range, it is crucial
to investigate how the performance measure is affected by the variation of
system parameters. In this paper, we demonstrate that such issue boils down to
the study of the variation of functions of uncertainty. Motivated by this
vision, we propose a general theory for inferring function of uncertainties. By
virtue of such theory, we investigate concentration phenomenon of random
vectors. We derive uniform exponential inequalities and multidimensional
probabilistic inequalities for random vectors, which are substantially tighter
as compared to existing ones. The probabilistic inequalities are applied to
investigate the performance of control systems with real parametric
uncertainty. It is demonstrated much more useful insights of control systems
can be obtained. Moreover, the probabilistic inequalities offer performance
analysis in a significantly less conservative way as compared to the classical
deterministic worst-case method.
