The law of the iterated logarithm for an associated random field. (English. Russian original) Zbl 1134.60025

Shashkin, A. P.

Russ. Math. Surv. 61, No. 2, 359-361 (2006); translation from Usp. Mat. Nauk 61, No. 2, 173-174 (2006).

This short note presents a law of the iterated logarithm (LIL) for wide-sense stationary centred and associated random fields together with an outline of the proof. The LIL holds for partial sums over cubes of positive integers increasing in each coordinate, provided that the following moment conditions hold. There should exist an absolute moment of higher order than second, uniformly over the field, and the decrease of covariances in the sense of Cox-Grimmett’s coefficient \(u_r\) should be at least of order \(O(r^{-\lambda})\) for some \(\lambda>0\) as \(r\) tends to infinity.

Reviewer: Peter Becker-Kern (Dortmund)

MSC:

60F15	Strong limit theorems
60G60	Random fields

Keywords:

law of the iterated logarithm; associated random field; wide-sense stationarity; Cox-Grimmett coefficient