Sanov property, generalized I-projection and a conditional limit theorem. (English) Zbl 0544.60011

Csiszár, Imre

Ann. Probab. 12, 768-793 (1984).

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)

Keywords:

asymptotic behaviour; empirical distribution; convergence in information; exponential family; conditionally asymptotically quasi-independent