Calder's Lemma

Calder’s Lemma is a mathematical statement used in probability and analysis, particularly involving martingales and convex functions. It asserts that under certain conditions, when applying a convex function to a sum of random variables (or martingales), the expected value (average) of the function’s output can be bounded or controlled by the sum of the expected values of the function applied to individual parts. In essence, it helps compare complex, combined processes to simpler, individual ones, ensuring that certain inequalities hold, which is useful in studying the behavior of stochastic systems and analyzing their stability or bounds.