Beyond Poisson: First-Passage Asymptotics of Renewal Shot Noise
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866917287337918464 |
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| author | Brémont, Julien |
| author_facet | Brémont, Julien |
| contents | 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 show analytically 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_21670 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Beyond Poisson: First-Passage Asymptotics of Renewal Shot Noise Brémont, Julien Statistical Mechanics Probability 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 show analytically 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. |
| title | Beyond Poisson: First-Passage Asymptotics of Renewal Shot Noise |
| topic | Statistical Mechanics Probability |
| url | https://arxiv.org/abs/2510.21670 |