Strong invariance principles for mixing random fields

Given a random field {ξν, ν∈Z+q} indexed by q-tuples of positive integers and satisfying a strong mixing condition we study the approximation of the partial sum field {Sn, n∈Z+q} by Brownian sheet. Setting {Mathematical expression} for 0<α<1 we show that in the domain Gα the approximation Sn - W (n) = O([n]1/2-λ) a.s. is possible where λ>0. We also construct an example showing that in a somewhat larger, similar type domain the above approximation is generally impossible, even with λ=0.