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 presents a comprehensive analytical framework for evaluating the autocovariance function of non‑Poisson dichotomous noise, focusing on a generalized random telegraph signal (RTS) whose waiting times follow a Mittag‑Leffler distribution. Traditional RTS models assume Poissonian switching events, leading to exponentially distributed waiting times, stationarity, and a simple Lorentzian power spectrum. However, many physical and biological systems exhibit heavy‑tailed waiting‑time statistics, rendering the standard approach inadequate.

The authors begin by reviewing the classic stationary RTS, where the probability of n switches in a time interval Δt follows a Poisson law and the autocovariance depends only on the lag Δt, yielding R_X X(Δt)=e^{−2|Δt|}. They then introduce renewal theory as a natural extension to handle arbitrary waiting‑time distributions. In a renewal process, event times t₀=0, t₁, t₂,… are generated by independent, identically distributed inter‑arrival times τ_i with density ψ(τ). The counting process N(t) and the residual (forward) waiting time g(y; t) are expressed through convolution relations (Eqs. 8–11).

To avoid cumbersome convolutions, the authors employ double Laplace transforms (with respect to the lag Δt and the observation time t). They derive a compact expression for the double‑Laplace transform of the probability P(n,Δt; t) of observing n renewals in the interval