## On the probability of large deviations in Banach spaces.(English)Zbl 0538.60008

Let $$\mu_ n$$, $$\mu$$ be a Borel probability measures on a real separable Banach space B. For $$a\in B$$, let $$h(a| \mu)=\sup \{f(a)-\log(\int \exp(f(x))\mu(dx)):\quad f\in B^*\}$$ and $$h(A| \mu)=\inf \{h(a| \mu):\quad a\in A\}.$$ The main results are expressed in the following theorems.
Theorem 1. If $$\{\mu_ n\}$$ converges weakly to $$\mu$$ and $$\sup_{n\geq 1}\int \exp(t\| x\| \mu_ n(dx)<\infty$$, $$\forall t>0$$, then the inequality $$\lim \sup_{n\to \infty}(1/n)\log \mu_ n^{*n}(nA)\leq - h(A| \mu)$$ holds for every closed subset $$A\subset B$$ and $$\lim \inf_{n\to \infty}(1/n)\log \mu_ n^{*n}(nA)\geq -h(A| \mu)$$ holds for every open subset $$A\subset B.$$
Theorem 2. If A is the closure of an open convex set which is flat at the point a and $$h(a| \mu)=h(A| \mu)$$ then $$\mu^{*n}(nA)\exp(nh(A| \mu))=O(1/\sqrt{n})$$.
Reviewer: A.J.Rachkauskas

### MSC:

 60B12 Limit theorems for vector-valued random variables (infinite-dimensional case) 60F10 Large deviations

### Keywords:

Banach space valued random variables
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