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

Bolthausen, Erwin

Ann. Probab. 12, 427-435 (1984).

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