Summary
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
for 0<α<1 we show that in the domain G α the approximation S n − 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.
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Berkes, I., Morrow, G.J. Strong invariance principles for mixing random fields. Z. Wahrscheinlichkeitstheorie verw Gebiete 57, 15–37 (1981). https://doi.org/10.1007/BF00533712
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DOI: https://doi.org/10.1007/BF00533712