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

2025/03/07 by Kolesko, Konrad, Meiners, Matthias, Tomic, Ivana · 1 citation
#45M05 #60J80 #60K15 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2503.05468

Abstract

We consider the Markov renewal equation F(t) = f(t) + \boldsymbolμ*F(t) for vector-valued functions f,F: ℝ → ℝ^p and a p × p matrix \boldsymbolμ of locally finite measures μ^i,j on [0,∞), i,j=1,…,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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• Convergence of complex martingales in supercritical multi-type general branching processes in L^q for 1 < q ≤ 2

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