= Density convergence of multiple weak-mixing correlations
{title2=$\operatorname{D-lim}_n\int f_0\prod_{j=1}^kU^{jn}f_j=\prod_{j=0}^k\int f_j$}
For real bounded observables, the <Cesaro limit> of the correlation comes from <arithmetic-progression multiple averages under weak mixing>. Apply the same result on the weakly mixing product system to $f_j\otimes f_j$ to obtain its second moment. The averaged squared error tends to zero, so the <mean-square criterion for convergence in density> gives the displayed density limit.
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