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On the lower bound of large deviation of random walks. (English) Zbl 0559.60031

Consider a random walk \(X_ 0,X_ 1,..\). on \({\mathbb{R}}^ n\) with resolvent R(x,dy) satisfying \(R(x,A)>0\) for every \(x\in {\mathbb{R}}^ n\) and open set \(A\subset {\mathbb{R}}^ n\). In the main theorem the author obtains a large deviation lower bound, i.e. \[ \lim \inf_{n\to \infty}(1/n) \log P_ x(L_ n(\omega,\cdot)\in N)\geq -I(\mu), \] where \(L_ n(\omega,\cdot)\) denotes the occupation measure, N is a weak neighborhood of \(\mu\) and I(\(\mu)\) is the Donsker-Varadhan functional. This extends an earlier result of the author dealing with compact state spaces.
Reviewer: J.Steinebach

MSC:

60F10 Large deviations
60J05 Discrete-time Markov processes on general state spaces
60G50 Sums of independent random variables; random walks
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