Almost sure growth of supercritical multi-type continuous state branching process

TL;DR

This paper proves that for finite-type supercritical multi-type continuous-state branching processes, the leading eigenvalue precisely determines the almost sure growth rate of each type, extending classical results to this continuous setting.

Contribution

It establishes that the leading eigenvalue of the process's linear semigroup exactly characterizes the almost sure growth rate for finite-type supercritical MCSBPs, aligning with classical Galton-Watson results.

Findings

The leading eigenvalue determines the almost sure growth rate.

The results extend classical multi-type Galton-Watson theory.

The growth rate matches the spectral radius of the linear semigroup.

Abstract

In Li (2011), Example 2.2, the notion of a multi-type continuous-state branching process (MCSBP) was introduced with a finite number of types, with the countably infinite case being proposed in Kyprianou and Palau (2017). One may consider such processes as a super-Markov chain on a countable state-space of types, which undertakes both local and non-local branching. In Kyprianou and Palau (2017) it was shown that, for MCSBPs, under mild conditions, there exists a lead eigenvalue which characterises the spectral radius of the linear semigroup associated to the process. Moreover, in a qualitative sense, the sign of this eigenvalue distinguishes between the cases where there is local extinction and exponential growth. In this paper, we continue in this vein and show that, when the number of types is finite, the lead eigenvalue gives the precise almost sure rate of growth of each type. This result matches perfectly classical analogues for multi-type Galton--Watson processes.