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. The main results rely on ergodicity requirements and an application of Nummelin splitting for continuous-time 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:强烈的不变原则描述了对随机过程部分总和进行布朗近距离近似的错误术语。 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, 我们获得了对广泛类别的ergodic Markov 过程的强烈不定原则 。 主要结果依赖于异端要求和对连续时间过程应用Nummelin 分离 。 强烈的偏差原则为分析依赖性结构环境中常用的静态差异估计数据提供了一个统一框架 。 我们展示了如何使用它来分析小滑铁滑的蒙特卡洛采样器模拟输出的批量方法 。 我们还利用强烈的近差结果对异性扩散的添加功能产生了波动效应 。