Pathwise Shrinkage and Probabilistic Calibration of the Multiplicative Half-Cauchy Process

Siddhesh Kulkarni

We characterize the multiplicative half-Cauchy process through its log-scale random-walk representation, establishing shrinkage, expansion, and critical regimes, almost-sure summability, local rebounds, and effective dimension. We also derive a semi-analytic calibration rule linking interpretable prior probability statements to its scale parameter.