Beyond Poisson: First-Passage Asymptotics of Renewal Shot Noise

TL;DR

This work derives the first universal asymptotic formula for the mean FPT $\langle T_b \rangle$ to reach level $b$ for renewal shot noise with general arrival statistics and exponential marks, and establishes a general framework for analyzing extreme events in non-Markovian systems with relaxation.

Abstract

The first-passage time (FPT) of a stochastic signal to a threshold is a fundamental observable across physics, biology, and finance. While renewal shot noise is a canonical model for such signals, analytical results for its FPT have remained confined to the Poisson (Markovian) case, despite the prevalence of non-Poisson arrival statistics in applications from neuronal spiking to gene expression. We break this long-standing barrier by deriving the first universal asymptotic formula for the mean FPT $\langle T_b \rangle$ to reach level $b$ for renewal shot noise with general arrival statistics and exponential marks. Our central result is a closed-form expression that reveals precisely how general inter-arrival statistics impact the naive Arrhenius law. We show that the short-time behavior of the interarrival distribution dictates universal scaling corrections, ranging from stretched-exponential to algebraic, that can dramatically accelerate threshold crossing. Furthermore, we argue and confirm numerically that the full FPT distribution becomes exponential at large thresholds, implying that $\langle T_b \rangle$ provides a complete asymptotic characterization. Our work, enabled by a novel exact solution for the moments of the noise, establishes a general framework for analyzing extreme events in non-Markovian systems with relaxation.

Figures (3)

• Figure 1: A typical realization of renewal shot noise $X(t)$ (solid line). The vertical green line signals the FPT $T_b$, where $X(t)$ exceeds the threshold $b=5$ (horizontal red line) for the first time.
• Figure 2: MFPT $\langle T_b\rangle$ of renewal shot noise with exponential marks (symbols: simulations; dashed lines: exact asymptotics, Eq. \ref{['mfpt-general']}). Interarrival times follow a Gamma distribution $w(t)=\tfrac{rk}{\Gamma(k)}(rk t)^{k-1}e^{-rk t}$ with mean rate $r$ and shape $k$, while marks are exponential with $\lambda=1$. Cases with $k=3$ (blue, $\gamma=1, r=0.4$) and $k=2$ (yellow, $\gamma=1.5, r=0.6$) highlight refractory effects ($w(0)=0$), whereas $k=0.75$ (green, $\gamma=4.5, r=1.6$) and the Poisson limit $k=1$ (red, $\gamma=2.5, r=1.2$) illustrate bursty dynamics. Statistical errors are smaller than the symbol size. The inset shows the simulated/theoretical ratio, confirming good quantitative agreement even at moderate $\lambda b$, with convergence speed set by $k$: larger $k$ suppresses short interarrivals and accelerates approach to the asymptotics.
• Figure 3: Numerical confirmation of the exponential distribution of $T_b$ for large $b$. The cumulative distribution function $\mathbb{P}(T_b < t)$ from simulations (in blue) is shown for a threshold $b=8$, with Gamma-distributed interarrivals (shape $k=2$, rate $r=0.6$), decay $\gamma=1.5$, and mark rate $\lambda=1$. The black dotted line is the expected exponential distribution $1-\exp(-t / \langle T_b \rangle)$, with $\langle T_b \rangle$ given by Eq. \ref{['mfpt-general']}. The excellent agreement strongly supports our assertion \ref{['exp-dist-tb']} that the FPT distribution is asymptotically exponential for large $b$.