Spectral Analysis and Long-Term Behavior of Discrete-Time Markov Chains

Abstract

This paper develops a spectral-probabilistic framework for examining the long-term dynamics of finite discrete-time Markov chains (DTMCs). The framework combines stochastic interpretation with eigenvalue-eigenvector analysis, explicit structural validation, probability propagation, and convergence diagnostics implemented in MATLAB. The principal four-state benchmark has transition matrix P and initial distribution x0 = [1, 0, 0, 0]. For the empirical tolerance ε = 10⁻3, the benchmark trajectory reaches stationarity within 13 steps, with stationary distribution π = (0.2313, 0.2916, 0.2676, 0.2094), subdominant spectral radius ρ = 0.5911, and spectral gap γ = 0.4089. The 13-step result is explicitly tied to the stated initial condition and tolerance. A formal worst-case calculation over all initial distributions gives the same 13-step threshold for this benchmark at ε = 10⁻3. A conservative spectral-gap reference scale is about 21 steps; because the benchmark is not assumed reversible, this value is reported as a comparison scale rather than as a direct theorem for the matrix. Additional synthetic matrices representing slow mixing, periodicity, and near-decomposability demonstrate that the diagnostic procedure distinguishes structurally different regimes. The study is therefore presented as a synthetic numerical benchmark rather than as a validated real-world reliability, deterioration, or risk model.