论文标题
无限记忆的随机链的最佳高斯浓度界限
Optimal Gaussian concentration bounds for stochastic chains of unbounded memory
论文作者
论文摘要
我们获得了最佳的高斯浓度界限（GCB），用于在可数字母上无限内存（SCUM）的随机链。这些随机过程也被称为“具有完整连接的链”或“ $ g $ measures”。我们考虑了内核上的两个不同条件：（1）当其振荡的总和小于一个或（2）当其变体的总和是有限的（即，即属于$ \ ell^1（\ Mathbb {n}）$时）。我们还将获得明确的常数作为模型参数的函数。证明基于最大耦合。从某种意义上说，我们的条件是最佳的，因为我们展示了没有GCB的浮渣示例，并且振荡的总和严格大于一个，或者该变化属于$ \ ell^{1+ε}（\ Mathbb {n}）$，对于任何$ε> 0 $。这些示例基于相变的存在。我们还将GCB的有效性扩展到一类函数，这些功能可能无限地取决于许多坐标。 我们通过三个应用程序说明了结果。首先，我们得出了Dvoretzky-kiefer-Wolfowitz型不平等现象，从而统一控制了经验度量的波动。其次，在有限的alphabet案例中，我们在$ \ bar {d} $ - 两个固定的浮渣之间获得了一个上限，并且作为副产品，我们在$ \ bar {d} $中获得了新的（显式）界限。第三，我们获得了特定类别可观察力的Birkhoff总和的指数收敛率。
We obtain optimal Gaussian concentration bounds (GCBs) for stochastic chains of unbounded memory (SCUMs) on countable alphabets. These stochastic processes are also known as "chains with complete connections" or "$g$-measures". We consider two different conditions on the kernel: (1) when the sum of its oscillations is less than one, or (2) when the sum of its variations is finite, i.e., belongs to $\ell^1(\mathbb{N})$. We also obtain explicit constants as functions of the parameters of the model. The proof is based on maximal coupling. Our conditions are optimal in the sense that we exhibit examples of SCUMs that do not have GCB and for which the sum of oscillations is strictly larger than one, or the variation belongs to $\ell^{1+ε}(\mathbb{N})$ for any $ε> 0$. These examples are based on the existence of phase transitions. We also extend the validity of GCB to a class of functions which can depend on infinitely many coordinates. We illustrate our results by three applications. First, we derive a Dvoretzky-Kiefer-Wolfowitz type inequality which gives a uniform control on the fluctuations of the empirical measure. Second, in the finite-alphabet case, we obtain an upper bound on the $\bar{d}$-distance between two stationary SCUMs and, as a by-product, we obtain new (explicit) bounds on the speed of Markovian approximation in $\bar{d}$. Third, we obtain exponential rate of convergence for Birkhoff sums of a certain class of observables.
