Large deviations for mixtures

Suppose the probability measures (mu(n)) on Theta obey a large deviation principle (LDP). Suppose too that mu(n) is concentrated on Theta(n) and that, for theta(n) is an element of Theta(n), with theta(n) --> theta is an element of Theta, the probability measures (P-theta(n)(n)) on X also obey an LDP. The main purpose of this paper is to give conditions which allow an LDP for the mixtures (P-n), given by P-n (A) = integral P-theta(n) (A) dmu(n) (theta), to be deduced. Chaganty (1997) also considered this question, but under stronger assumptions. The treatment here follows that of Dinwoodie and Zabell (1992) who, motivated by exchangeability, considered the ease where mu(n) does not vary with n.