ConvexAnalysisProbabilityTheory
This inequality should be viewed as a generalization of the convexity condition:
where . In other words, the function lies below the line between any two points on the function, where this condition could hold either globally or locally, depending on if we restrict the points . The generalization to Jensen’s holds by induction on the sums above:
where and . Probabilistically, Jensen’s is a continuous generalization of the above. If is a r.v. such that :
Jensen’s naturally extends to conditional expectations and can be used to define a submartingale from a martingale :