What is the mathematical approach for determining the limit of predictability

Question

Some physical phenomena, like turbulence, wind-driven ocean waves, and ship motion, can be demonstrated to be "random" in the long period. However, for short periods, the phenomena can be predicted accurately. My question is, for how long can a phenomena be predicted with a specified degree of accuracy. I've been trying to answer the question by trial and error, trying different models and evaluating how well the model does over a wide range of conditions. The results are favorable, but now there is the more general question of proving the limits of predictability. There's a similar question regarding the robustness of the predictions.