  For a given pair of positive integers $d$ and $N$ with $N \geq 2$, for
strictly stationary random fields that are indexed by the $d$-dimensional
integer lattice and satisfy $N$-tuplewise independence, the dependence
coefficients associated with the $\rho$-, $\rho'$-, and $\rho^*$-mixing
conditions can decay together at an arbitrary rate. Another, closely related
result is also established. In particular, these constructions provide classes
of examples pertinent to limit theory for random fields that involve such
mixing conditions together with certain types of "extra" assumptions on the
marginal and bivariate (or $N$-variate) distributions.
