# zbMATH — the first resource for mathematics

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