Stability of weak mixing under powers and products ($T^r,T\times T$)

Every positive power of a weak mixing transformation is weakly mixing: its nonnegative discrepancies are a subsequence whose average is bounded by $r$ times the full average. Products are weakly mixing because correlations of simple tensors factor; the absolute product discrepancy is bounded by the sum of bounded multiples of the single-system discrepancies. Density of finite tensor sums gives the full $L^2$ assertion.