Transition phenomena for many-dimensional renewal equation of special kind (Q2777847)

The author considers the many-dimensional renewal equation \(X^{\varepsilon}(t)= A^{\varepsilon}(t)+\int_0^t K^{\varepsilon}(du)X^{\varepsilon}(t-u)\). The case of a scalar renewal equation was considered by \textit{D. S. Silvestrov} [Theory Probab. Math. Stat. 20, 113-130 (1980); translation from Teor. Veroyatn. Mat. Stat. 20, 97-117 (1979; Zbl 0431.60084)]. The asymptotic behaviour of solutions of many-dimensional renewal equations was investigated by \textit{V. M. Shurenkov} [Math. USSR, Sb. 40, 107-123 (1981); translation from Mat. Sb., n. Ser. 112(154), 115-132 (1980; Zbl 0434.60090)] under the assumption that the matrix \(K[0,\infty)=\lim_{\varepsilon\to 0}K^{\varepsilon}[0,\infty)\) is an irreducible one. In this paper the asymptotic properties of the renewal matrix \(H^{\varepsilon}(t)=\sum_{n=0}^{\infty}(K^{\varepsilon}(t))^{(\ast n)}\) as \(\varepsilon\to 0\) and \(t\to\infty\) are investigated under assumptions that the measure matrix \(K^{\varepsilon}(\cdot)\) converges weakly to \(K(\cdot)\) as \(\varepsilon\to 0\) and \(K[0,\infty)=\lim_{\varepsilon\to 0}K^{\varepsilon}[0,\infty)\) is a reducible block-diagonal matrix.