In this paper, we investigate a coupling Markov chain for the VKF method, where an excessively long run of the chain is stopped if the number of its steps exceeds the total length of the trajectories preliminarily computed for the chain. For this version of the algorithm, a submultiplicativity lemma is proved. In addition, a theorem on the probability with which the length of the trajectory exceeds a certain threshold is proved by reasoning a la Fekete’s lemma. Finally, we prove that the probabilities of the results yielded by the ordinary and stopped coupling Markov chains differ in terms of the total variation distance by an exponentially small quantity with respect to the number of preliminary runs.

中文翻译：

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VKF方法的可乘性和耦合马尔可夫链的终止

在本文中，我们研究了用于VKF方法的耦合马尔可夫链，其中，如果步数超过了为该链初步计算出的轨迹的总长度，则该链的超长运行将停止。对于此算法版本，证明了一个可乘性引理。另外，通过推导la Fekete引理证明了关于轨迹长度超过某个阈值的概率的一个定理。最后，我们证明了普通和停止耦合的马尔可夫链所产生的结果的概率在总变化距离方面相对于初步运行的数量以指数形式变小。