A Proof of Walsh's Convergence Theorem Using Couplings

Walsh has recently proved the norm convergence of all nonconventional ergodic averages corresponding to polynomial sequences in discrete nilpotent acting groups. He deduced this convergence from an equivalent "finitary" assertion of stability over arbitrarily long-time intervals for these averages, which is proved by essentially finitary means. The present paper shows how the induction at the heart of Walsh's proof can also be implemented using more classical notions of ergodic theory, such as couplings and characteristic factors.