A Gaussian-Remainder Hierarchy for Sums of Random Variables with Big-Jump Statistics

What the math gives to ML

The paper provides a constructive Gaussian-remainder hierarchy for sums of independent finite-variance variables whose tails remain non-Gaussian at finite sample sizes. Its key transferable mechanism is the first-order convolution approximation, which preserves the Gaussian center while retaining a single-big-jump correction in the crossover and tail regions. A practical neural-network use is a tail-aware minibatch optimizer that estimates the distribution of per-example gradient projections, predicts aggregate-gradient tail risk, and adapts the learning rate or clipping threshold before rare large updates destabilize training.