Moderate, large and super large deviations principles for Poisson process with uniform catastrophes

TL;DR

The paper addresses the deviation behavior of Poisson processes with uniform catastrophes under three scaling regimes. It generalizes prior LLN/LDP results by introducing a scaling function $\,(T)$ and proving LDPs across sublinear (MDP), linear, and superlinear (SLDP) regimes, with corresponding rate functions $I_1$, $J_k$, and $I_2$. The analysis hinges on decomposing the process into a regular growth part and a catastrophes component, employing exponential bounds, truncation of catastrophe sizes, and coupling arguments to establish both upper and lower bounds for deviations. The results provide a complete large deviations framework for this class of processes, with implications for population dynamics and related stochastic models where abrupt catastrophes interact with regular growth.

Abstract

In this paper, we expand and generalize the findings presented in our previous work on the law of large numbers and the large deviation principle for Poisson processes with uniform catastrophes. We study three distinct scalings: sublinear (moderate deviations), linear (large deviations), and superlinear (superlarge deviations). Across these scales, we establish different yet coherent rate functions.

Theorems & Definitions (16)

• definition 1
• definition 2
• definition 3
• theorem 1: LRL
• theorem 2: LDP
• theorem 3: MDP
• theorem 4: SLDP
• lemma 1
• proof
• lemma 2: LYL