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Sanov property, generalized I-projection and a conditional limit theorem. (English) Zbl 0544.60011
Known results on the asymptotic behaviour of the probability that the empirical distribution $$\hat P_ n$$ of an i.i.d. sample $$X_ 1,...,X_ n$$ with common distribution $$P_ X$$ belongs to a given convex set $$\Pi$$ of probability measures, and new results on that of the joint distribution of $$X_ 1,...,X_ n$$ under the condition $$\hat P_ n\in \Pi$$ are obtained simultaneously, using an information-theoretic identity. Related results available in the literature concern the case when the condition $$\hat P_ n\in \Pi$$ represents a finite number of constraints on sample means; then, under various regularity hypotheses, the convergence of $$P_{X| \hat P_ n\in \Pi}$$ to the I-projection of $$P_ X$$ on $$\Pi$$ has been established.
These results are generalized in four directions: (i) more general sets $$\Pi$$ are concerned; (ii) the I-projection of $$P_ X$$ on $$\Pi$$ need not exist (but the generalized I-projection, $$P^*$$, exists); (iii) a stronger kind of convergence $$P_{X| \hat P_ n\in \Pi}\to P^*$$ is established (convergence in information); (iv) r.v.s. $$X_ 1,...,X_ n$$ under the condition $$\hat P_ n\in \Pi$$ are shown to behave like i.i.d. r.v.s. with common distribution $$P^*$$ (it is described in an exact way by the notion of asymptotic quasi-independence introduced in the paper). When $$\hat P_ n\in \Pi$$ is the event that the sample mean of a V-valued statistic $$\psi$$ is in a given convex subset of V, a locally convex topological vector space, the limiting conditional distribution of (either) $$X_ i$$ is characterized as a member of the exponential family determined by $$\psi$$ through the unconditional distribution $$P_ X$$, while $$X_ 1,...,X_ n$$ are conditionally asymptotically quasi-independent.
Reviewer: B.Kryžienė

##### MSC:
 60B10 Convergence of probability measures 60F10 Large deviations 60B12 Limit theorems for vector-valued random variables (infinite-dimensional case) 62B10 Statistical aspects of information-theoretic topics 94A17 Measures of information, entropy 82B05 Classical equilibrium statistical mechanics (general)
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