Mean ergodic convergence from near-diagonal subadditivity

Abstract

We prove a mean ergodic theorem whose subadditivity hypothesis is required only for nearly equal block lengths. A positive unital integral-preserving operator on a probability space need not be ergodic or invertible. Our main mechanism is conditional: convergence of the invariant projections of uniformly integrable normalized inputs, together with both ordered inequalities at length differences at most two, forces convergence in L^1. Combining this mechanism with an existing scalar shrinking-corridor theorem gives invariant L^p convergence, for every finite p≥1, when the relative corridor width has divergent square integral and eventually admits differences two. Nonnegative inputs require only individual L^p membership. For signed inputs, uniform integrability of the normalized negative-part pth powers is necessary and sufficient. The scalar width threshold and its converse are inherited; the contribution is their Markov mean-ergodic lift. Under the same scalar-admissible width and norm criterion, signed integrable inputs also satisfy lim sup g_n/n=F almost surely, with a possibly nonconstant invariant limit. The upper control is uniform over macroscopic operator powers and lengths. An all-legal adjacent-halving tree has a vanishing occupation gradient; a classical maximal-inequality argument admits integrable rewards directly. No uniform boundedness assumption is needed.