Beyond Poisson: First-Passage Asymptotics of Renewal Shot Noise

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, even though non-Poisson arrival statistics are common in systems from neuronal spiking to gene expression. Here, we overcome this long-standing limitation by deriving a universal asymptotic formula for the mean FPT ⟨T_{b}⟩ to reach level b for renewal shot noise with arbitrary arrival statistics and exponential marks. Our central result is a simple, closed-form expression that exposes the physical mechanism by which temporal correlations in arrivals modulate the baseline Arrhenius law. We show that bursty arrivals introduce universal scaling corrections that markedly accelerate threshold crossings. In turn, nonbursty arrivals remain Arrhenius-like, directly linking temporal burstiness to Arrhenius scaling. Furthermore, we show and confirm numerically that the full FPT distribution becomes exponential at large thresholds, implying that ⟨T_{b}⟩ provides a complete asymptotic characterization. Our Letter, enabled by a novel exact expression for the moments of the noise, establishes a general framework for analyzing extreme events in non-Markovian systems with relaxation.