A random telegraph signal of Mittag-Leffler type

A general method is presented to explicitly compute autocovariance functions for non-Poisson dichotomous noise based on renewal theory. The method is specialized to a random telegraph signal of Mittag-Leffler type. Analytical predictions are compared to Monte Carlo simulations. Non-Poisson dichotomous noise is non-stationary and standard spectral methods fail to describe it properly as they assume stationarity.

💡 Research Summary

The paper addresses a fundamental gap in the statistical description of dichotomous (two‑state) noise whose switching times do not follow the exponential law of a Poisson process. Traditional random telegraph signals (RTS) assume exponentially distributed waiting times, which guarantees Markovian dynamics and stationarity. In many physical, biological, and engineered systems, however, the residence‑time statistics exhibit heavy‑tailed behavior, rendering the Poisson assumption invalid and causing the signal to become non‑stationary. Standard spectral techniques, which rely on a time‑invariant autocorrelation function, therefore fail to capture the true dynamics.

To overcome this limitation, the authors develop a general method based on renewal theory. They consider a dichotomous process X(t) that takes values ±1 and switches whenever a renewal event occurs. The key quantity is the number of renewals N(t) up to time t. By expressing the autocovariance as

 C(t, τ)=⟨X(t)X(t+τ)⟩=⟨(−1)^{N(t+τ)−N(t)}⟩,

the problem reduces to evaluating the statistics of N(t) for an arbitrary waiting‑time distribution ψ(τ). Using the renewal equation and Laplace transforms, the authors obtain closed‑form expressions for the mean and variance of N(t) and, consequently, for C(t, τ).

The method is then specialized to a waiting‑time distribution of Mittag‑Leffler type, ψα(τ)=−d/dτ Eα