Sharp Orlicz Endpoints for Spatial-Temporal Ergodic Averaging

What the math gives to ML

The paper provides a transferable local-stability principle for compositions of temporal and spatial averaging: pointwise temporal convergence becomes stable under shrinking spatial observation windows when the temporal maximal function has an integrable majorant. It also identifies sharp Orlicz thresholds, with regular-time averages requiring L log^(q+1) L and arbitrary sequences requiring L log^q L at normalization N Lambda_q(N). A neural-network analogue is to train representations or predictors whose temporal trajectories have controlled Orlicz moments and whose finite-horizon maximal fluctuation is bounded across local input neighborhoods. This gives a concrete robustness mechanism and predicts a logarithmic integrability transition rather than merely adding generic smoothness.