Predictive inference takes the sequence of one-step-ahead predictive distributions as the primitive object for learning and inference, rather than an explicit model- prior specification. This approach naturally encompasses Bayesian procedures, but also applies to prediction--based learning rules that are only asymptotically exchangeable or arise from computationally motivated approximations.

We study asymptotic inference induced by predictive learning rules that are not necessarily exchangeable but converge almost surely to a random limiting distribution. Our main contribution is a functional Doob--type theorem for predictive inference. We show that, under suitable regularity conditions, the conditional distribution of the limiting predictive process, centered at the current predictive distribution and suitably rescaled, converges almost surely to a Gaussian law in an appropriate functional space. The associated covariance structure is explicitly characterized in terms of predictive updates, yielding an analytic approximation of the implicit posterior distribution and providing a direct tool for uncertainty quantification and predictive efficiency assessment.

Under i.i.d. observations, we obtain a Bernstein--von Mises theorem for the predictive distribution, showing asymptotic normality of the implicit posterior centered at the predictive mean, with variance determined by the learning dynamics of the predictive rule.

Finally, we discuss extensions of the framework to supervised settings with regressors, where predictive distributions depend on covariates. In this context, central limit theorems for predictive distributions with fixed covariate values provide Gaussian approximations for conditional laws, with applications to regression and modern prediction-based learning methods.