The functional law of the iterated logarithm for associated random fields

There are a number of interesting models in mathematical statistics, reliability theory and statistical physics described by means of families of associated random variables. In particular, any collection of independent real-valued random variables is automatically associated. The goal of the paper is to provide simply verifiable conditions to guarantee the validity of the functional law of the iterated logarithm for real-valued associated random field $\left\{X_j,\,j\in\mathbb Z^d\right\}$ defined on the lattice $\mathbb Z^d$, $d\geq1$. If this field is wide-sense stationary, the mentioned conditions read: $\sup_{j}E|X_j|^s<\infty$ for some $s\in(2,3]$ and the estimate $u(n)=O(n^{-\lambda})$ as $n\to\infty$ for some $\lambda >d/(s-1)$ is admitted by the Cox–Grimmett coefficient $u(n)$ having an elementary expression in terms of the covariance function of the field. Being based on the new maximal inequality established by A. V. Bulinski and M. S. Keane, the proof employs the methods of the known papers by V. Strassen, J. Chover and I. Berkes. An essential role is played also by the estimates of the convergence rates in the central limit theorem for associated random fields obtained in the author's recent publications. The paper is organized as follows: § 1 is the introduction describing the association concept and indicating the investigations in the domain of limit theorems for families of associated variables. Some necessary notations and the formulation of the main result are contained in § 2. The functional law of the iterated logarithm is proved in § 3 with the help of 6 lemmas.