马尔可夫链Hoeffding不等式的一个简短算子证明
A short operator proof of Hoeffding inequalities for Markov chains
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中文总结 AI 辅助
本文用两个关键观察（算子分解与秩一扰动）给出马尔可夫链加性泛函及随机矩阵Hoeffding不等式的简短算子证明，恢复并改进现有结果。
中文摘要 AI 辅助
我们给出一个简短的算子理论证明，针对一般状态空间上具有$L^2(\pi)$谱隙的马尔可夫链的加性泛函，证明了尖锐的Hoeffding不等式，恢复了Fan等人（2021）的界。我们进一步表明，同样的论证可得到Neeman等人（2024）关于马尔可夫相关随机矩阵的Hoeffding不等式。该证明基于两个关键观察。首先，转移算子通过León-Perron算子$Q_\lambda$分解为$P=Q_\lambda^{1/2} K Q_\lambda^{1/2}$，其中$\\|K\\|=1$。这将矩生成函数界定为一步算子范数的乘积。其次，每个一步算子是乘法算子的秩一扰动，因此界定其范数归结为验证一个标量预解条件。标量凸性和经典Hoeffding引理随后给出所需估计。这绕过了现有证明中使用的渐近累积量生成函数、本质谱分析和极值两态比较。在矩阵情形中，多矩阵Golden-Thompson不等式首先将矩生成函数归结为提升Hilbert空间上的算子问题。随后同样的两步适用：到常数的投影变为有限秩，且Schur补论证产生相同的标量估计。Neeman等人（2024）的矩阵Hoeffding不等式随之得出。对于复Hermitian加项，直接在复Hilbert空间上工作将界改进因子$2$。
英文摘要
We give a short operator-theoretic proof of the sharp Hoeffding inequality for additive functionals of a Markov chain on a general state space with an $L^2(π)$ spectral gap, recovering the bound of Fan et al. (2021). We further show that the same argument yields the Hoeffding inequality of Neeman et al. (2024) for Markov-dependent random matrices. The proof rests on two key observations. First, the transition operator factors as $P=Q_λ^{1/2} K Q_λ^{1/2}$ through the León-Perron operator $Q_λ$, with $\|K\|=1$. This bounds the moment generating function by a product of one-step operator norms. Second, each one-step operator is a rank-one perturbation of a multiplication operator, so bounding its norm reduces to verifying a scalar resolvent condition. Scalar convexity and the classical Hoeffding lemma then give the desired estimate. This bypasses the asymptotic cumulant generating function, essential-spectrum analysis, and extremal two-state comparison used in existing proofs. In the matrix setting, the multi-matrix Golden-Thompson inequality first reduces the moment generating function to an operator problem on a lifted Hilbert space. The same two steps then apply: the projection onto constants becomes finite-rank, and a Schur-complement argument yields the same scalar estimate. The matrix Hoeffding inequality of Neeman et al. (2024) follows. For complex Hermitian summands, working directly on the complex Hilbert space improves the bound by a factor of $2$.
发表机构
- MaLGa center, Dipartimento di Matematica, Università degli Studi di Genova（马耳他数学中心，热那亚大学数学系）
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