Exponential moment for the extinction time

Prove that the extinction time of the multitype Markovian branching process with type-1 population-size-dependent reproductive parameters possesses a finite exponential moment, thereby establishing exponential ergodicity and, under irreducibility on the non-absorbing states, supporting the existence of a quasi-stationary distribution.

Background

The paper proves that extinction occurs almost surely and that the extinction time has finite moments of every polynomial order. However, the moment bounds obtained for the first-passage time grow like a constant multiple of $(r!)^2$, which is insufficient to imply the existence of a finite exponential moment.

A finite exponential moment is identified as a necessary condition for the existence of a quasi-stationary distribution. In the irreducible case, the cited results would also make this condition sufficient. Thus, establishing an exponential moment would strengthen the paper's extinction result and potentially yield quasi-stationary behavior for the branching process.