Large deviations for a general class of random vectors. (English) Zbl 0534.60026

Ellis, Richard S.

Ann. Probab. 12, 1-12 (1984).

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.

Reviewer: J.Steinebach

MSC:

60F10	Large deviations
26A51	Convexity of real functions in one variable, generalizations

Keywords:

convexity; entropy function; exponential convergence