In \cite{P69}, Walter Philipp wrote that ``\dots the law of the iterated logarithm holds for any process for which the Borel-Cantelli Lemma, the central limit theorem with a reasonably good remainder and a certain maximal inequality are valid.'' Many authors \cite{DW80}, \cite{I68}, \cite{N91}, \cite{OY71}, \cite{Y79} have followed this plan in proving the law of the iterated logarithm for sequences (or fields) of dependent random variables. We carry on this tradition by proving the law of the iterated logarithm for a random field whose correlations satisfy an exponential decay condition like the one obtained by Spohn \cite{Sp86} for certain Gibbs measures. These do not fall into the $\phi$-mixing or strong mixing cases established in the literature, but are needed for our investigations \cite{SS01} into diffusions on configuration space. The proofs are all obtained by patching together standard results from \cite{OY71}, \cite{Y79} while keeping a careful eye on the correlations.

Let $L_p=L_p(X,\mu)$, $1\leq p\leq\infty$, be the usual Banach Spaces of real valued functions on a complete non-atomic probability space. Let $(T_1,\ldots,T_{K})$ be $L_2$-contractions. Let $0<\varepsilon < \delta\leq1$. Call a function $f$ a $\delta$-spanning function if $\|f\|_2 = 1$ and if $\|T_kf-Q_{k-1}T_kf\|_2\geq\delta$ for each $k=1,\ldots,K$, where $Q_0=0$ and $Q_k$ is the orthogonal projection on the subspace spanned by $(T_1f,\ldots,T_kf)$. Call a function $h$ a $(\delta,\varepsilon)$-sweeping function if $\|h\|_\infty\leq1$, $\|h\|_1<\varepsilon$, and if $\max_{1\leq k\leq K}|T_kh|>\delta-\varepsilon$ on a set of measure greater than $1-\varepsilon$. The following is the main technical result, which is obtained by elementary estimates. There is an integer $K=K(\varepsilon,\delta)\geq1$ such that if $f$ is a $\delta$-spanning function, and if the joint distribution of $(f,T_1f,\ldots,T_Kf)$ is normal, then $h=\bigl((f\wedge M)\vee(-M)\bigr)/M$ is a $(\delta,\varepsilon)$-sweeping function, for some $M>0$. Furthermore, if $T_k$s are the averages of operators induced by the iterates of a measure preserving ergodic transformation, then a similar result is true without requiring that the joint distribution is normal. This gives the following theorem on a sequence $(T_i)$ of these averages. Assume that for each $K\geq1$ there is a subsequence $(T_{i_1},\ldots,T_{i_K})$ of length $K$, and a $\delta$-spanning function $f_K$ for this subsequence. Then for each $\varepsilon>0$ there is a function $h$, $0\leq h\leq1$, $\|h\|_1<\varepsilon$, such that $\limsup_iT_ih\geq\delta$ a.e.. Another application of the main result gives a refinement of a part of Bourgain's ``Entropy Theorem'', resulting in a different, self contained proof of that theorem.