Large deviations for a general class of random vectors
Consider a general sequence \(\{Y_ n\}_{n=1,2,...}\) of random vectors taking values in \(R^ d\) or in a certain infinite-dimensional locally convex Hausdorff topological space \({\mathcal Z}\). Suppose that \(c(t)=\lim_{n\to \infty}a_ n^{-1}\log E_ n\{<t,Y_ n>\}\) exists for all \(t\in R^ d\) or \(t\in {\mathcal T}\), the topological dual of \({\mathcal Z}\). Then, given c with certain properties, upper and lower large deviation estimates are obtained, i.e. \(\lim \sup_{n\to \infty}a_ n^{-1}\log Q_ n(K)\leq -I(K)\) for closed (or compact) subsets K, and \(\lim \inf_{n\to \infty}a_ n^{-1}\log Q_ n(G)\geq -I(G)\) for open subsets G, where \(Q_ n\) denotes the distribution of \(a_ n^{-1}Y_ n\) and I is the entropy function defined via the Legendre-Fenchel transform of c. An application to finite-state Markov chains is discussed as well as the notion of exponential stochastic convergence, which is closely related to the underlying large deviation property of the sequence under consideration.
scientific article; zbMATH DE number 3848316