Ergodicity and regime recoverability in finite Markov-modulated random walks

We study a finite Markov-modulated random walk (MMRW) on a one-dimensional lattice with reflecting boundaries, where a hidden finite-state Markov environment Eₜ selects at each time step a regime-specific random walk kernel for the position process Xₜ. Treating the joint process Zₜ= (Xₜ, Eₜ) as a Markov chain on the product state space, we construct its transition kernel and derive a discrete-time Master Equation. We then state simple structural assumptions on the environment kernel and the regime-specific spatial kernels that guarantee irreducibility and aperiodicity of the joint chain; since the state space is finite, standard Markov chain theory implies existence and uniqueness of a stationary distribution and convergence in total variation. On top of this stochastic model we formulate supervised learning problems on simulated trajectories. Using sliding windows of increments Xₜ, we build summary-statistic feature vectors and study when standard classifiers can infer the hidden environment state from the observed path. For a two-regime MMRW, support vector machines, random forests and gradient boosting all recover the environment with test accuracy and macro F1 close to 0. 9. We then consider short-horizon prediction of Xₓ+₁ and compare three gradient-boosting predictors: a regime-agnostic baseline, a regime-aware model that uses environment probabilities estimated from Task A, and an oracle that observes the true environment. In the default single-run configuration, the regime-aware model yields only a modest downstream effect relative to the baseline, while the oracle quantifies the performance level that would be reachable under perfect regime information. Finally, we use the decoded environment sequence to estimate the environment transition matrix and its stationary distribution, and compare these to the corresponding quantities from the true environment, including a Frobenius-norm distance and spectral gaps. This links the probabilistic structure of the MMRW with the practical ability of standard machine-learning methods to recover latent regimes and coarse environment dynamics from finite data.