Explosion speed of continuous state branching processes indexed by the Esscher transform

TL;DR

The paper tackles explosion in CSBPs by coupling a non-conservative process $Z$ with a family of conservative CSBPs $Z^{(\varepsilon)}$ through Esscher-transformed branching mechanisms $\varphi^{(\varepsilon)}(\lambda)=\varphi(\lambda+\varepsilon)-\varphi(\varepsilon)$, and proves $Z^{(\varepsilon)}\to Z$ a.s. as $\varepsilon\downarrow0$ via a shared probability space and Lamperti-type path construction. It then analyzes the explosion speed by examining first passage times $\sigma_{k/h(n)}^{(n)}$ of $Z^{(n)}$ to levels $k/h(n)$, identifying a trichotomy of convergence regimes via the class $\mathcal{Z}$ and a limiting random variable $\zeta$ possibly shifted by a constant $c(h)$ in the weak limit, especially under regularly varying branching mechanisms. The results provide necessary and sufficient conditions for convergence in distribution and $L^1$, and offer a sufficient, verifiable criterion for almost sure convergence, revealing a residual mass phenomenon in the weak limit on a critical domain. Collectively, the work yields a unified, a.s.-coupled framework for studying explosion speed in CSBPs indexed by the Esscher transform and connects fine asymptotics to the Lamperti representation of CSBPs.

Abstract

A branching process $Z$ is said to be non conservative if it hits $\infty$ in a finite time with positive probability. It is well known that this happens if and only if the branching mechanism $\varphi$ of $Z$ satisfies $\int_{0+}dλ/|\varphi(λ)|<\infty$. We construct on the same probability space a family of conservative continuous state branching processes $Z^{(\varepsilon)}$, $\varepsilon\ge0$, each process $Z^{(\varepsilon)}$ having $\varphi^{(\varepsilon)}(λ)=\varphi(λ+\varepsilon)-\varphi(\varepsilon)$ as branching mechanism, and such that the family $Z^{(\varepsilon)}$, $\varepsilon\ge0$ converges a.s.~to $Z$, as $\varepsilon\rightarrow0$. Then we study the speed of convergence of $Z^{(\varepsilon)}$, when $\varepsilon\rightarrow0$, referred to here as the explosion speed. More specifically, we characterize the functions $f$ with $\lim_{\varepsilon\rightarrow0} f(\varepsilon)=\infty$ and such that the first passage times $σ_\varepsilon=\inf\{t:Z^{(\varepsilon)}_t\ge f(\varepsilon)\}$ converge toward the explosion time of $Z$. Necessary and sufficient conditions are obtained for the weak convergence and convergence in $L^1$. Then we give a sufficient condition for the almost sure convergence.

Figures (1)

• Figure 1: $Z$ in red, $Z^{(n)}$ in orange and $Z^{(m)}$ in blue for $n>m$.

Theorems & Definitions (44)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 3.1
• Remark 3.1
• Proposition 3.1
• Proposition 3.2
• Proposition 3.3
• Proposition 3.4