We study true self-avoiding walk (TSAW) as a mechanism for improving empirical integral estimation via Markov chain Monte Carlo (MCMC). We consider finite-state adaptive sampling dynamics associated with an irreducible Markov kernel $P$ on a finite set, with stationary distribution $π$, in which the transition probabilities are penalized according to empirical overuse. Our main result is that the empirical occupation counts $L_t(i)$ and transition counts $N_t(i,j)$ of the resulting TSAW-based walk satisfy \[ L_t(i)-tπ_i = O(\sqrt{\log t}) \quad\text{and}\quad N_t(i,j)-tπ_iP_{ij}=O(\sqrt{\log t}) \qquad\text{almost surely} \] for every state $i$ and every edge $(i,j)$ with $P_{ij}>0$. Consequently, for every bounded function $f:V\to\mathbb R$, the error of our integral estimator converges as \[ \left|\frac1t\sum_{s=0}^{t-1} f(X_s)-\sum_{i\in V}π_i f(i)\right| = O\left(\frac{\sqrt{\log t}}{t}\right) \qquad\text{almost surely}. \] These results show that, in contrast with the usual $t^{-1/2}$ error scaling for empirical averages under standard random-walk-based methods, TSAW-based estimator yields empirical integral errors of order $O(\sqrt{\log t}/t)$ almost surely, thereby achieving a substantially sharper dependence on the sample size $t$.

翻译：我们研究了真正自回避行走（TSAW）作为一种通过马尔可夫链蒙特卡罗（MCMC）改进经验积分估计的机制。考虑在有限集上与不可约马尔可夫核 $P$ 相关的有限状态自适应采样动力学，其平稳分布为 $π$，其中转移概率根据经验过度使用而受到惩罚。我们的主要结果是，由此产生的基于TSAW的行走的经验占据计数 $L_t(i)$ 和转移计数 $N_t(i,j)$ 几乎必然地满足：对于每个状态 $i$ 和每条满足 $P_{ij}>0$ 的边 $(i,j)$，有 \[ L_t(i)-tπ_i = O(\sqrt{\log t}) \quad\text{和}\quad N_t(i,j)-tπ_iP_{ij}=O(\sqrt{\log t}). \] 因此，对于每个有界函数 $f:V\to\mathbb R$，我们的积分估计器的误差几乎必然地收敛为 \[ \left|\frac1t\sum_{s=0}^{t-1} f(X_s)-\sum_{i\in V}π_i f(i)\right| = O\left(\frac{\sqrt{\log t}}{t}\right). \] 这些结果表明，与标准随机游走方法下经验平均的通常 $t^{-1/2}$ 误差标度相比，基于TSAW的估计器几乎必然地产生阶为 $O(\sqrt{\log t}/t)$ 的经验积分误差，从而实现了对样本量 $t$ 显著更优的依赖关系。