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Asymptotic expansions of solutions to Markov renewal equations and their application to general branching processes
Konrad Kolesko, Matthias Meiners, Ivana Tomic · 2025-03-07 · via math.PR updates on arXiv.org

We consider the Markov renewal equation $F(t) = f(t) + \boldsymbolμ*F(t)$ for vector-valued functions $f,F: \mathbb{R} \to \mathbb{R}^{p}$ and a $p \times p$ matrix $\boldsymbolμ$ of locally finite measures $μ^{i,j}$ on $[0,\infty)$, $i,j=1,\ldots,p$. Sgibnev [Semimultiplicative estimates for the solution of the multidimensional renewal equation. {\em Izv.\ Ross.\ Akad.\ Nauk Ser.\ Mat.}, 66(3):159--174, 2002] derived an asymptotic expansion for the solution $F$ to the above equation. We give a new, more elementary proof of Sgibnev's result, which also covers the reducible case. As a corollary, we infer an asymptotic expansion for the mean of a multi-type general branching process with finite type space counted with random characteristic. Finally, some examples are discussed that illustrate phenomena of multi-type branching.

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