Infinite-color randomly reinforced urns with dominant colors

Prediction-based uncertainty quantification for exchangeable sequences

“…We study the limit behaviour of the normalized difference √ n( P(A) − P n (A)) in terms of a stronger form of convergence, namely, almost sure convergence of the conditional distributions, or more briefly, a.s. conditional convergence (see [29]). This kind of convergence is studied, in different contexts, by Fortini & Petrone [15], Berti et al [30,31], Crimaldi [32] and Sariev et al [33]. Theorem 3.1 below proves that, under fairly mild assumptions, √ n( P(A) − P n (A)) converges in the sense of a.s. conditional convergence to a Normal distribution with mean zero and positive variance U A (there defined); that is (3.3).…”

“…). A novelty of our approach, as in [15,33], is in the statistical use we make of this kind of convergence, as informative of the asymptotic form of the posterior distribution of the unknown P(A). Indeed, we will read (3.3) as asymptotically implying that…”

“…In the probabilistic literature, limit results of this kind aim at studying the convergence of the conditional probability Pnfalse(Afalse)false(x1:nfalse) to its limit Pfalse~false(Afalse)false(x1,x2,…false). A novelty of our approach, as in [15,33], is in the statistical use we make of this kind of convergence, as informative of the asymptotic form of the posterior distribution of the unknown Pfalse~false(Afalse). Indeed, we will read (3.3) as asymptotically implying that Pfalse~false(Afalse)∣x1:n≈N(Pn(A),UAn),with P-probability one.…”