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

الأصل · EN

For sub-additive ergodic processes {Xₘ,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 Var X₀,ₙ n²χ, 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 Var X₀,ₙ = O(n/(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.