Nonlinear Markov Chains (nMC) are regarded as the original (linear) Markov Chains with nonlinear small perturbations. It fits real-world data better, but its associated properties are difficult to describe. A new approach is proposed to analyze the ergodicity and even estimate the convergence bounds of nMC, which is more precise than existing results. In the new method, Coupling Markov about homogeneous Markov chains is applied to reconstitute the relationship between distribution at any times and the limiting distribution. The convergence bounds can be provided by the transition probability matrix of Coupling Markov. Moreover, a new volatility called TV Volatility can be calculated through the convergence bounds, wavelet analysis and Gaussian HMM. It's tested to estimate the volatility of two securities (TSLA and AMC). The results show TV Volatility can reflect the magnitude of the change of square returns in a period wonderfully.

翻译：非线性 Markov 链条(nMC) 被视为原始的(线性) Markov 链条, 有非线性的小扰动。 它更适合真实世界的数据, 但相关的属性很难描述 。 提议了一种新的方法来分析正统性, 甚至估计nMC 的趋同界限, 这比现有结果更为精确 。 在新的方法中, Coupling Markov 条条条条条被套用来重建任何时间的分布和限制分布之间的关系 。 Coupling Markov 的过渡概率矩阵可以提供趋同界限 。 此外, 一种称为 TVVVolatility 的新的波动可以通过趋同线、 波 分析 和 Gausian HMM 来计算 。 它被测试来估计两种证券( TSLA 和 AMC ) 的波动性。 结果显示 TVolatility 能够完美地反映一段时期平方回报变化的程度 。