Sanov Property, Generalized I-Projection and a Conditional Limit Theorem

Known results on the asymptotic behavior of the probability that the empirical distribution Pₙ of an i.i.d. sample X₁, ⋯, Xₙ belongs to a given convex set Π of probability measures, and new results on that of the joint distribution of X₁, ⋯, Xₙ under the condition Pₙ ∈ Π are obtained simultaneously, using an information-theoretic identity. The main theorem involves the concept of asymptotic quasi-independence introduced in the paper. In the particular case when Pₙ ∈ Π is the event that the sample mean of a V-valued statistic ψ is in a given convex subset of V, a locally convex topological vector space, the limiting conditional distribution of (either) Xᵢ is characterized as a member of the exponential family determined by ψ through the unconditional distribution PX, while X₁, ⋯, Xₙ are conditionally asymptotically quasi-independent.