Density convergence of multiple weak-mixing correlations (${D-lim}_n\int f_0{\prod }_{j=1}^k{U}^{jn}{f}_j={\prod }_{j=0}^k\int f_j$)

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.