\displaystyle \begin{aligned} \frac 1n \int (S_n f)^2\,d\mu &= \frac 1n \int \Big(\sum_{i=0}^{n-1} f\circ T^i\Big) \Big( \sum_{j=0}^{n-1} f\circ T^j\Big)\,d\mu \\ &= \frac 1n \sum_{i=0}^{n-1} \sum_{k=-i}^{n-1-i} \int f\cdot (f\circ T^k)\,d\mu \\ &= \sum_{k=-n+1}^{n-1} \frac{n-1-|k|}{n} \int f\cdot (f\circ T^k)\,d\mu \end{aligned}