Asymptotic expansions of solutions to Markov renewal equations and their application to general branching processes

We consider the Markov renewal equation $F(t) = f(t) + \boldsymbol{\mu}*F(t)$ for vector-valued functions $f,F: \mathbb{R} \to \mathbb{R}^{p}$ and a $p \times p$ matrix $\boldsymbol{\mu}$ of locally finite measures $\mu^{i,j}$ 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.