A Central Limit Theorem and Law of the Iterated Logarithm for a Random Field with Exponential Decay of Correlations

摘要

In cite{P69}, Walter Philipp wrote that ``dots the law of theiterated logarithm holds for any process for which the Borel-CantelliLemma, the central limit theorem with a reasonably good remainder anda certain maximal inequality are valid.'' Many authors cite{DW80},cite{I68}, cite{N91}, cite{OY71}, cite{Y79} have followed thisplan in proving the law of the iterated logarithm for sequences (orfields) of dependent random variables.We carry on this tradition by proving the law of the iteratedlogarithm for a random field whose correlations satisfy an exponentialdecay condition like the one obtained by Spohn cite{Sp86} forcertain Gibbs measures. These do not fall into the $phi$-mixing orstrong mixing cases established in the literature, but are needed forour investigations cite{SS01} into diffusions on configurationspace. The proofs are all obtained by patching together standard results fromcite{OY71}, cite{Y79} while keeping a careful eye on thecorrelations.