The telegraph process alternates between two states ±a at Poisson rates λ₊, λ₋. Explore exact power spectral density (Lorentzian), integrated displacement statistics, and driven systems with colored noise.
Telegraph process: 2-state CTMC with rates λ₊ (from −a to +a) and λ₋ (from +a to −a). Stationary: p(+a)=λ₊/(λ₊+λ₋), p(−a)=λ₋/(λ₊+λ₋). Mean=a(λ₊-λ₋)/(λ₊+λ₋). Autocorrelation: C(τ)=σ²·exp(-|τ|(λ₊+λ₋)). PSD: Lorentzian S(ω)=2σ²λ/(λ²+ω²) where λ=λ₊+λ₋. Integral of telegraph is non-Gaussian despite CLT; transition regime visible. Applications: ion channel gating, gene expression bursting, two-level systems.