Telegraph Process: Two-State Switching

Switching Rates

Rate ON→OFF (λ): 1.0 s⁻¹ Rate OFF→ON (μ): 2.0 s⁻¹ High state level: 1.0 Low state level: 0.0

Telegraph Process

λ: ON→OFF rate (dwell time ~ Exp(λ))
μ: OFF→ON rate (dwell time ~ Exp(μ))

Stationary dist: P(ON)=μ/(λ+μ)
Mean = μ·hi + λ·lo)/(λ+μ)
Variance = λμ(hi-lo)²/(λ+μ)²

Power spectral density (Lorentzian):
S(f) ∝ 1/[(λ+μ)² + (2πf)²]

Diffusion limit: Brownian motion

The telegraph process is the canonical two-state noise model. In gene expression, promoter switching between active/inactive states creates transcriptional bursts. In neuroscience, ion channels toggle between open and closed. The Lorentzian power spectrum is its fingerprint — a flat (white) plateau below the corner frequency λ+μ.