Rate of convergence of the mean for sub-additive ergodic sequences

For sub-additive ergodic processes {X_m,n} with weak dependence, we analyze the rate of convergence of EX0,n/n to its limit g. We define an exponent γ given roughly by EX0,n~ng+nγ, and, assuming existence of a fluctuation exponent χ that gives VarX_0,n~n^2χ, we provide a lower bound for γ of the form γ≥χ. The main requirement is that χ≠1/2. In the case χ=1/2 and under the assumption VarX_0,n=O(n/(log n)^β) for some β>0, we prove γ≥χ-c(β) for a β-dependent constant c(β). These results show in particular that non-diffusive fluctuations are associated to non-trivial γ. Various models, including first-passage percolation, directed polymers, the minimum of a branching random walk and bin packing, fall into our general framework, and the results apply assuming χ exists. In the case of first-passage percolation in Zd, we provide a version of γ≥-1/2 without assuming existence of χ.