Strong invariance principles describe the error term of a Brownian approximation of the partial sums of a stochastic process. While these strong approximation results have many applications, the results for continuous-time settings have been limited. In this paper, we obtain strong invariance principles for a broad class of ergodic Markov processes. Strong invariance principles provide a unified framework for analysing commonly used estimators of the asymptotic variance in settings with a dependence structure. We demonstrate how this can be used to analyse the batch means method for simulation output of Piecewise Deterministic Monte Carlo samplers. We also derive a fluctuation result for additive functionals of ergodic diffusions using our strong approximation results.

翻译：强烈的变差原则描述了布朗近似值对随机过程部分总和的错误术语。 虽然这些强烈近近近结果有许多应用, 但持续时间设置的结果有限。 在本文中, 我们为一系列广泛的ergodic Markov 过程获得了强烈的变差原则。 强烈的变差原则为分析依赖性结构环境中常用的无现性差异估计值提供了一个统一框架。 我们展示了如何使用这一框架来分析微小的确定性蒙特卡洛采样员模拟输出的批量手段方法。 我们还利用强烈的近差结果对循环扩散的添加功能产生了波动效应。