First we polish an argument of Remillard and Theodorescu for the weak convergence of the empirical probability generating process. Then we prove a general inequality between probability generating processes and the corresponding empirical processes, which readily implies a rate of convergence and trivializes the problem of weak convergence: whenever the empirical process or its non-parametric bootstrap version, or the parametric estimated empirical process or its bootstrap version converges, so does the corresponding probability generating process. Derivatives of the generating process are also considered.
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