次可加遍历序列均值的收敛速率
Rate of convergence of the mean for sub-additive ergodic sequences
摘要：
对于弱相依的次可加遍历过程$\{X_{m,n}\}$，分析了$\mathbb{E}X_{0,n}/n$到其极限$g$的收敛速率.我们定义一个指数$γ$大致由$\mathbb{E}X_{0,n} \sim ng + n^γ$，并假设存在一个波动指数$χ$，给出$\mathrm{Var}~X_{0,n} \sim n^{2χ}$，我们提供了一个下界$γ$的形式$γ\geq χ$。主要要求是$χ\neq 1/2$。在$χ=1/2$的情况下，在对某些$β>0$有$\mathrm{Var}~X_{0,n} = O(n/(\log n)^β)$的假设下，我们证明了对$β$依赖的常数$c(β)$有$γ\geq χ- c(β)$.这些结果表明，特别是非扩散波动与非平凡的$γ$。各种模型，包括第一次通过渗流，定向聚合物，最小的分支随机行走和装箱，落入我们的一般框架，并假设$χ$存在的结果适用。在$\mathbb Z^d$中的第一通道渗流的情况下，我们提供了一个版本的$γ\geq -1/2$，而不假设存在$χ$。
英文摘要：
For sub-additive ergodic processes $\{X_{m,n}\}$ with weak dependence, we analyze the rate of convergence of $\mathbb{E}X_{0,n}/n$ to its limit $g$. We define an exponent $γ$ given roughly by $\mathbb{E}X_{0,n} \sim ng + n^γ$, and, assuming existence of a fluctuation exponent $χ$ that gives $\mathrm{Var}~X_{0,n} \sim n^{2χ}$, we provide a lower bound for $γ$ of the form $γ\geq χ$. The main requirement is that $χ\neq 1/2$. In the case $χ=1/2$ and under the assumption $\mathrm{Var}~X_{0,n} = O(n/(\log n)^β)$ for some $β>0$, we prove $γ\geq χ- c(β)$ 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 $\mathbb Z^d$, we provide a version of $γ\geq -1/2$ without assuming existence of $χ$.
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