Inequalities between event and time averages

We consider a stochastic process with an embedded point process in a statio nary and ergodic context. Under a "lack of anticipation" assumption for the evolution of the process vis-a-vis the point process, a new better (worse) than used expectation property for the point process, and a monotonicity a ssumption for the behavior of the process between points, inequalities betw een event and time averages are obtained. Sample path monotonicity between points is not required (as is the case with existing approaches) and can be replaced with a simple monotonicity requirement for the expected value of the process between points. Inequalities between conditional event and time averages are also examined via a novel argument involving a conditional ve rsion of the Palm inversion formula.