On large deviations for uniformly strong mixing sequences (Q1198593)

Let \({\mathbf V}\) be a Hausdorff locally convex topological vector space, \({\mathbf X}\subset {\mathbf V}\) be convex and compact in the relative topology of \({\mathbf V}\). Let \(\{X_ n\}_{n\in Z}\) be a stationary sequence of bounded \({\mathbf X}\)-valued random variables satisfying \(\phi\)-mixing condition with fast enough decreasing coefficients. The most general result of this paper states that \(Z_ n=(X_ 1+\dots+X_ n)/n\), \(n\geq 1\), satisfies the large deviation principle, i.e., there is a convex lower semicontinuous function \(I:{\mathbf X}\to[0,\infty)\) with compact level sets and such that \[ \limsup_{n\to\infty}n^{-1}\log P(Z_ n\in A)\leq-\inf_{x\in A}I(x) \] for each closed set \(A\subset{\mathbf X}\) and \[ \liminf_{n\to\infty}n^{-1}\log P(Z_ n\in A)\geq-\inf_{x\in A}I(x) \] for each open set \(A\subset{\mathbf X}\). Moreover, \[ \lim_{n\to\infty}n^{-1}\log E\exp\{n\lambda(Z_ n)\}=L(\lambda) \] exists for each \(\lambda\in{\mathbf V}^*\) and the rate function is given by \(I(x)=\sup\{\lambda(x)-L(\lambda)\): \(\lambda\in{\mathbf V}^*\}\). Similar results in the case \({\mathbf V}={\mathbf R}^ d\) or for the sequence of empirical measures are obtained as well as in the case of \(\psi\)-mixing stationary random sequences with values in a separable Banach space.