Proportional-limit asymptotics for Diaconis-Ylvisaker-penalised logistic regression with fitted intercept

This paper develops estimator-level asymptotic theory for maximum Diaconis-Ylvisaker prior penalised likelihood for logistic regression with a jointly fitted intercept and nonzero prior slope direction in the proportional-limit regime. For $\mathrm{N}(\mathbf{0}_p, p^{-1}\mathbf{I}_p)$ Gaussian covariates and $p/n\toκ\in(0,1)$, a conditional convex Gaussian min--max analysis yields almost-sure convergence of the fitted intercept and a pseudo-Lipschitz empirical law for the slope estimator. This estimator-level law gives asymptotic limits for out-of-sample scores, classification error, optimal thresholding and oracle calibration. It also yields oracle-adjusted fixed-block $Z$-statistics under isotropic Gaussian covariates, and yields the main ingredient in establishing the limiting distribution of the penalised likelihood-ratio test statistic and identifies the rescaling to recover the nominal chi-square distribution. We extend these results to Gaussian designs with arbitrary deterministic mean and positive-definite covariance via affine centering and whitening and discuss extensions to subgaussian covariates. Finally, we propose a consistent response-moment estimator of the oracle parameters entering the state equations that govern the slope limiting law and are required for feasible inference.