The Law of the Iterated Logarithm for Sums of Exponentially Stabilizing Functionals

We consider sums of exponentially stabilizing functionals (introduced by Penrose and Yukich) of Poisson point processes in $d$-dimensional Euclidean space. The asymptotic behavior of such sums is studied in terms of a random field (defined on the lattice $\mathbb Z^d$) each element of which is a certain sum of functionals with respect to the corresponding unit cube in $\mathbb R^d$. For this random field, we obtain an exponential estimate of the decrease of the strong mixing coefficient and establish the law of the iterated logarithm.