Strong invariance principles for mixing random fields

SummaryGiven 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 $$G_\alpha = \{ (n_1 ,...,n_q ) \in Z_ + ^q :n_k \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \geqslant } (\mathop \Pi \limits_{1\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } i\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } q,i \pm k} n_i )^\alpha , k = 1,...,q\}$$ 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.