Kappa distributions as asymptotic marginals of exponential family ensembles

What the math gives to ML

The paper gives a constructive mechanism by which inverse-gamma fluctuations of total kinetic energy generate kappa-distributed single-particle velocities in the large-particle limit. The transferable asset is an analytically controlled heavy-tailed noise source: sampling one inverse-gamma scale and Gaussian directions produces a Student-t or kappa perturbation with a tunable tail exponent and a Gaussian limit as the spectral index grows. In neural-network training, this can be used as a scale-mixture SGD noise schedule or as a heavy-tailed posterior sampler, with a sharp moment boundary that predicts when gradient-noise variance and kurtosis exist. The most direct experiment is to compare this noise against Gaussian SGD while sweeping the kappa parameter across the finite-variance and finite-fourth-moment thresholds.