[Paper Review] The maximal jump and local convergence of continuous-state branching processes

Xin He, Zenghu Li|arXiv (Cornell University)|Dec 15, 2014

Stochastic processes and statistical mechanics11 references3 citations

TL;DR

This paper investigates the distribution of the maximal jump and local convergence properties of continuous-state branching processes (CB processes) under various conditioning mechanisms. Using stochastic equations and Lévy-Itô decomposition, it derives exact expressions and asymptotics for local and global maximal jumps, establishes absolute continuity between the maximal jump distribution and the Lévy measure, and proves local convergence to a CBI process under conditioning on large maximal jump, large width, or large total mass—extending classical results on large height conditioning.

ABSTRACT

We study the distribution of the maximal jump of continuous-state branching processes. Several exact expressions and explicit asymptotics of both the local maximal jump and the global maximal jump are obtained. We also compare the distribution of the maximal jump and the Lévy measure to get several absolute continuity results. Then we study local convergence of continuous-state branching processes under various conditionings. We obtain complete results under the conditioning of large maximal jump, and partial results under two other conditionings, which are, the conditioning of large width, and, the conditioning of large total mass.

Motivation & Objective

• To characterize the distribution of the maximal jump in continuous-state branching processes, both locally and globally.
• To establish absolute continuity relations between the maximal jump distribution and the Lévy measure in critical, subcritical, and supercritical cases.
• To extend classical local convergence results for CB processes beyond the standard large height conditioning.
• To investigate local convergence under three alternative conditionings: large maximal jump, large width, and large total mass.
• To provide a unified framework for understanding the scaling limits of CB processes under diverse pathwise constraints.

Proposed method

• Uses stochastic equations and the Lévy-Itô decomposition to analyze jump dynamics in CB processes.
• Derives the distribution of the local maximal jump via an ODE involving the Lévy measure and the branching mechanism.
• Applies excursion representation under excursion measure to obtain results applicable to Lévy trees.
• Employs time-changed processes and exponential tilting to analyze conditional distributions under large jump and large mass conditioning.
• Utilizes the Markov property and asymptotic analysis of the Laplace exponent to derive convergence limits.
• Applies conditional equivalence results via exponential martingales to transform conditioning measures.

Experimental results

Research questions

• RQ1 How does the distribution of the maximal jump in a CB process relate to its Lévy measure?
• RQ2 Under what conditions is the distribution of the global maximal jump absolutely continuous with respect to the Lévy measure?
• RQ3 What is the limiting behavior of a CB process when conditioned to have a large maximal jump?
• RQ4 How does local convergence under large maximal jump compare to classical large height conditioning?
• RQ5 What are the limiting distributions under conditioning on large width or large total mass?

Key findings

• The tail of the local maximal jump and the tail of the Lévy measure are asymptotically of the same order, as shown in Theorem 3.3.
• In the subcritical case, the global maximal jump's tail is asymptotically equivalent to the Lévy measure's tail, as established in Theorem 3.7.
• The distribution of the global maximal jump is absolutely continuous with respect to the Lévy measure in the critical and subcritical cases, per Theorem 3.11.
• For the local maximal jump, absolute continuity with the Lévy measure holds in all cases, as proven in Theorem 3.14.
• Under large maximal jump conditioning, the local limit is a CBI process with immigration, identical to the classical large height limit in the critical case, as shown in Theorem 4.2.
• Under large total mass conditioning, the paper conjectures convergence to a CBI process with modified branching and immigration mechanisms, depending on the criticality and existence of critical tilting parameters.