  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.
